Harmonic Bergman spaces, Hardy-type spaces and harmonic ... · Bergman spaces and applications to...

Universita degli Studi di Milano-Bicocca

Dottorato di Ricerca in Matematica Pura e Applicata

Harmonic Bergman spaces, Hardy-type spaces

and harmonic analysis of a symmetric diffusion

semigroup on Rn

Francesca Salogni

Advisor:

Professor Stefano Meda

Contents

Introduction i

I Hardy-type spaces 1

1 Generalized Bergman spaces in Rn 3

1.1 Mean value properties for polyharmonic functions . . . . . . . . . . . . . . . 3

1.2 Generalized Bergman spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 The reproducing kernel of generalized Bergman spaces . . . . . . . . . . . . 12

1.3.1 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.4 Estimates for generalized Bergman kernels . . . . . . . . . . . . . . . . . . . 23

1.5 Application to Hardy spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2 Bergman and Hardy spaces on Riemannian manifolds 37

2.1 Basic definitions and background material . . . . . . . . . . . . . . . . . . . 37

2.2 Harmonic Bergman spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.3 Hardy-type spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.4 Calderon–Zygmund decomposition and interpolation . . . . . . . . . . . . . 56

2.5 Spectral multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

2.6 Doubling property and scaled Poincare inequality . . . . . . . . . . . . . . . 67

II A symmetric diffusion semigroup 73

3 The semigroup generated by the operator A 75

3.1 Background material and preliminary results . . . . . . . . . . . . . . . . . . 75

3.2 Local Calderon–Zygmund theory . . . . . . . . . . . . . . . . . . . . . . . . 85

3.3 The maximal operator for the semigroup Ht . . . . . . . . . . . . . . . . . . 91

3

3.4 Multiplier operators for A . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

Bibliography 99

Aknowledgements

I would like to thank my advisor Stefano Meda for his guidance, support and friendship.

I also thank Maria Vallarino and Andrea Carbonaro for their sympathy and patience.

Finally, I want to express my gratitude to all the colleagues at the University of Milano-

Bicocca: I really thank you for making these years so special.

5

Introduction

This thesis is divided into two parts, which deal with quite diverse subjects.

The first part is, in turn, divided into two chapters. The first focuses on the development

of new function spaces in Rn, called generalized Bergman spaces, and on their application

to the Hardy space H1(Rn). The second is devoted to the theory of Bergman spaces on

noncompact Riemannian manifolds which possess the doubling property and to its relation-

ships with spaces of Hardy type. The latter are tailored to produce endpoint estimates for

interesting operators, mainly related to the Laplace–Beltrami operator.

The second part is devoted to the study of some interesting properties of the operator

A f = −1

2∆f − x· ∇f ∀f ∈ C∞c (Rn),

which is essentially self-adjoint with respect to the measure

dγ−1(x) = πn/2 e|x|2

dλ(x) ∀x ∈ Rn,

where λ denotes the Lebesgue measure, and of the semigroup that A generates.

The generalized Bergman spaces in Rn and their applications to H1(Rn) are discussed

in Chapter 1, their extension to Riemannian manifolds is described in Chapter 2, and the

analysis of the operator A occupies Chapter 3. Here we briefly describe the main results we

have obtained, and illustrate the relationships with related results in the literature.

i

ii

Bergman spaces and applications to Hardy spaces

The Euclidean case

Suppose that p is a number satisfying 1 ≤ p <∞, and let Ω be an open bounded set in Rn.

The harmonic Bergman space bp(Ω) is the space of all harmonic functions u on Ω such that

‖u‖bp :=( ∫

Ω

|u|p dλ)1/p

<∞. (0.0.1)

The analogous space of holomorphic functions was introduced and studied by S. Bergman

[Be]. The Bergman spaces of holomorphic functions have been and still are the object

of intensive studies in harmonic analysis of functions of several complex variables and in a

variety of related contexts. The interest in harmonic Bergman spaces is comparatively recent

(see, for instance, [ABR, KK, CKY] and the refences therein). One source of interest lies in

the fact that the harmonic Bergman kernel, i.e., the reproducing kernel of b2(Ω), is related

to the Green function for the bi-harmonic Laplace equation on Ω (see, for instance, [Ma] and

the references therein).

For k ≥ 1 we define the generalized harmonic Bergman space bpk(Ω) to be the space of all

functions satisfying (0.0.1) that are k-harmonic, i.e., such that

∆ku = 0 in Ω

in the sense of distributions (hence in the classical sense, by elliptic regularity). In Chapter 1

we establish some basic properties of b2k(B), where B is a ball of Rn. In particular, we find

an orthonormal basis of B2k(B) consisting of k-harmonic polynomials of b2

k(B) and compute

the reproducing kernel (or Bergman kernel) of b2k(B). The latter is the unique function Rk

B

on B ×B such that

u(x) =

∫B

RkB(x, y)u(y) dy ∀x ∈ B ∀u ∈ b2

k(B).

A closed formula for R1B is known (see [ABR, Theorem 8.13]), and estimates for derivatives

of R1B may be easily deduced from it. In fact, a closed formula for Rk

B is also known for

k > n/2 [Ma]. This formula is obtained by first computing explicitly the Green function

for ∆k and then deducing the exact formula for RkB. This procedure is interesting in itself,

but gives a formula for RkB only for large values of k (compared to the dimension n) and it

is a little bit involved. Our approach works for every n and k, and it is based on a simple

iii

trick, which relates RkB to the extended Poisson kernel on B (see (1.4.1) below), and on the

behaviour of the L2-norms of certain k-harmonic polynomials αkj : j ∈ N as j tends to

infinity. In fact, we first compute the exact values of these L2-norms, which are solutions

to certain linear systems, and then deduce their asymptotic behaviour as j tends to infinity.

The major drawback of our approach is that we have not found a direct and elegant way

to find these solutions. Anyway, apart from this, estimates for RkB and its derivatives follow

quite easily from our formula.

The theory of k-harmonic Bergman spaces we developed has an interesting application

to the theory of the Hardy space H1(Rn). Recall that H1(Rn) is a natural substitute for the

Lebesgue space L1(Rn) in many problems arising in real and harmonic analysis. The seminal

works of C. Fefferman and E.M. Stein [FeS], R.R. Coifman [Co] and R.H. Latter [La] pro-

vide many different characterisations of H1(Rn). In particular, the atomic characterisation,

proved in one dimension by Coifman and in higher dimensions by Latter, opened the way

to generalisations of the theory of Hardy spaces to more general settings, such as spaces of

homogeneous type in the sense of Coifman and G. Weiss [CW].

These are metric measured spaces (X, ρ, µ), where ρ is a (pseudo-) distance on X and µ

is a Borel measure satisfying some mild assumptions and possessing the doubling property.

Recall that µ is doubling if there exists a constant D such that

µ(2B) ≤ Dµ(B) for every metric ball B in X. (0.0.2)

Here 2B is the ball with the same centre as B and twice the radius.

We briefly recall the definition of the Hardy space H1(X) [CW].

A function a in L2(X) with support contained in a metric ball B is said to be a (1, 2)-atom

if it satisfies the following size and cancellation conditions

(i)(∫

B|a|2 dµ

)1/2 ≤ µ(B)−1/2,

(ii)∫Xa dµ = 0.

A function f in L1(X) belongs to H1(X) if it admits a decomposition of the form

f =∑j

cj aj,

where cj is in `1 and the aj are (1, 2)-atoms. Coifman and Weiss proved that we obtain

the same space if we consider (1, p)-atoms instead of (1, 2)-atoms. Here p is in (1,∞], and a

iv

(1, p)-atom is defined much as an (1, 2)-atom, but with the size condition (i) above replaced

by the following

(i)’(∫

B|a|p dµ

)1/p ≤ µ(B)−1/p′ , where p′ denotes the index conjugate to p, and the integral

must be suitably interpreted when p =∞.

There are interesting variants of the atomic decomposition of H1(Rn) in the literature. In-

deed, as a consequence of the maximal characterisation of H1(Rn) (see, for instance, [St2,

Chapter III]), it is known that we may consider (1, p)-atoms with more than one vanishing

moment. More precisely, for every nonnegative integer N , denote by PN the finite dimen-

sional space of polynomials of degree at most N in Rn. Then we still get H1(Rn) if we

consider functions in L1(Rn) which admit an atomic decomposition in terms of (1, p)-atoms

satisfying the following modified cancellation condition

(ii)’∫Bjaj(x) q(x) dx = 0 ∀q ∈PN .

An application of the theory of k-harmonic Bergman spaces developed in Chapter 1 is to

prove that functions in H1(Rn) admit an atomic decomposition in terms of atoms satisfying

suitable infinite dimensional cancellation conditions. This turns out to have interesting links

with the theory of k-harmonic functions defined on balls of Rn.

Suppose that k is a positive integer, p is in (1,∞) and B is a ball. Define A pk (B) to be

as the space of all functions A in Lp(B) satisfying the following conditions

(i)(∫

B|A(x)|p dx

)1/p ≤ |B|−1/p′ ;

(ii)∫BA(x) q(x) dx = 0 for all k-harmonic polynomials q.

The collection of all functions belonging to A pk (B) for some ball B will be denoted by A p

k

and its elements will be called Xk,p-atoms. Note that the cancellation condition (ii) above

may be equivalently expressed as “orthogonality” to the k-harmonic Bergman spaces bp′

k (B)

(here p′ denotes the index conjugate to p).

In Theorem 1.5.7 we shall give an elementary proof that every function in H1(Rn) indeed

admits an atomic decomposition in terms of Xk,p-atoms.

The case of Riemannian manifolds

In Chapter 2, we consider the case of connected complete noncompact Riemannian man-

ifolds M , which possess the doubling property of balls (see (0.0.2)) and satisfy a relative

v

Faber–Krahn inequality (see (2.1.2)). Recall that these properties are equivalent to a on-

diagonal upper estimate for the heat kernel of the type

(DUE) ht(x, x) ≤ C

µ(B(x,

√t)) ∀x ∈M ∀t > 0.

Recently, Hardy-type spaces associated to operators have been introduced in various settings

(see [R, AMR, MMV1, MMV2, HLMMY] and the references therein).

T. Coulhon and X.T. Duong [CD] proved that under the above assumptions on M , the

Riesz transform f 7→∣∣∇L −1/2

∣∣ is of weak type (1, 1). An interpolation argument with the

trivial L2(M) boundedness of the Riesz transform gives the Lp(M) boundedness for all p

in (1, 2). P. Auscher, A. McIntosh and E. Russ [AMR] proved that the Riesz transform is

bounded from a Hardy-type space H1d∗(Λ

0T ∗M) to L1(M). It is quite a difficult task to

prove that this Hardy-type space agrees with ours. A proof of this equivalence is implicit in

[HLMMY].

In the special case where M supports a scaled Poincare inequality (see (2.6.1)) Russ [Ru]

proved that the Riesz transform is bounded from H1(M) to L1(M). Moreover, H1(M) =

H1d∗(Λ

0T ∗M) (see [AMR]), hence the Riesz transform is bounded from H1d∗(Λ

0T ∗M) to

L1(M). We give a different proof of the fact that these two Hardy spaces agree, based on De

Giorgi’s regularity theorem for elliptic equations, which, in turn, in the setting of Riemannian

manifolds is a well-known consequence of an important result obtained independently by

Grigor’yan and Saloff-Coste (see [Sa1]).

It is fair to say that, although some of the results contained in this chapter are already

present in the literature, our point of view is, to the best of our knowledge, new. Furthermore,

we believe that our approach is somewhat simpler, and that it may be adapted to other

situations where the doubling condition fails (see, for instance, [MMV2, MMV4]).

The operator A

This part of the thesis is dedicated to the analysis of the operator A , defined by

A f = −1

2∆f − x· ∇f ∀f ∈ C∞c (Rn).

For every real number β, denote by γβ both the function

γβ(x) = π−nβ/2 e−β|x|2

∀x ∈ Rn, (0.0.3)

vi

and the measure on Rn whose density with respect to the Lebesgue measure λ is γβ. In

particular, γ1 is the Gauss measure on Rn. Denote by Qγβ the Dirichlet form, defined by

Qβ(f) =1

2

∫Rn|∇f(x)|2 dγβ(x) ∀f ∈ C∞c (Rn).

A simple integration by parts shows that A is the operator associated to Q−1, and that the

Ornstein–Uhlenbeck operator

L f = −1

2∆f + x· ∇f ∀f ∈ C∞c (Rn)

is the operator associated to Q1. The Ornstein–Uhlenbeck operator generates a diffu-

sion semigroup, which has been the object of many investigations during the last two

decades. In particular, efforts have been made to study operators related to the Ornstein–

Uhlenbeck semigroup, with emphasis on maximal operators (see [MPS, GMMST2] and

the references therein), Riesz transforms [MUu, GMST1, MMS] and functional calculus

[GMST2, GMMST1, HMM].

The purpose of this part of the thesis is to investigate the analogues for the operator A

of some of the results contained in the aforementioned papers. In many cases, the methods

developed for L go through for A , and the proofs of the results for A are straightforward

modifications of the proofs of the corresponding results for A . There are exceptions, however,

as in the determination of the region of holomorphy in Lp(γ−1) of the semigroup generated

by A .

We note that the operators A and L are unitarily equivalent, so that the analysis of A

on L2(γ−1) is equivalent to the analysis of L on L2(γ1). There is no such equivalence as far

as analysis on Lp is concerned. Therefore, although there are analogies between the analysis

of A (and of related operators) on Lp(γ−1) and that of L (and of related operators) on

Lp(γ1), the results for A are not directly deducible from those for L .

Note also that the measure γ−1 is infinite, whereas γ1 is a probability measure. Further-

more, if B(x, r) denotes the Euclidean ball with centre x and radius r, then γ−1

(B(x, r)

)grows more than exponentially with r, as r tends to infinity. An interesting line of re-

search in analysis on Riemannian manifolds aims at understanding the behaviour of the

Laplace–Beltrami operator on a Riemannian manifold and of certain related operators (spec-

tral multipliers, Riesz transforms, . . . ) under certain geometric assumptions. Quite often

these concern the volume growth and the curvature of the manifold. Many investigations

vii

have been made in the case where the volume growth of the manifold is polynomial, or at

most exponential, but there are virtually no results in the case of superexponential growth.

Though A is not the Laplace–Beltrami operator of any Riemannian metric on Rn, it is

the “radial part” of the Laplace–Beltrami operator of a Riemannian metric on a suitable

warped product of Rn and Tn. We hope that some of the results contained in Chapter 3 will

give some indications for further investigations on manifolds with superexponential volume

growth.

Altogether, we believe that the result presented here will be valuable for any further

investigation of A .

Here is a summary of the results contained in Chapter 3. In Section 3.1, we find an explicit

formula for the semigroup Htt≥0 generated by A , and we shall prove that Htt≥0 is a

symmetric diffusion semigroup on (Rn, γ−1).

A well known result by V.A. Liskevich and M.A. Perelmuter [LP] states that each sym-

metric diffusion semigroup Ttt≥0 on (X,µ) extends to a bounded holomorphic semigroup

on Lp(µ) with angle at least φp = arccos(2/p−1), i.e., the map t→ Tt extends to an analytic

Lp-operator-valued function z → Tz defined on the sector Sφp = z ∈ C : |arg(z)| < φp,such that |||Tz|||Lp(µ) ≤ 1 for each z ∈ Sφp . It is known that the angle φp in the theorem

of Liskevich and Perelmuter is optimal. Indeed, a celebrated theorem of J.B. Epperson [E,

Theorem 1.1] asserts that the semigroup Mz generated by the Ornstein–Uhlenbeck operator

extends to a bounded operator on Lp(γ) if and only if z belongs to the set

Ep = x+ iy ∈ C : |sin y| ≤ tanφp sinhx,

which contains the sector Sφp and is tangent to the rays e±iφpR+ at the origin. We shall

prove that the same result holds for the analytic continuation Hz of the semigroup Ht.In Section 3.2, we develop the theory of local Calderon–Zygmund operators.

In Section 3.3 we consider the maximal operator associated to the semigroup Ht,defined by

H ∗f = supt>0|Htf | . (0.0.4)

By the Banach principle (see [G]), it is well known that the study of the boundedness of

H ∗ on Lp(γ−1) is related to the problem of the almost everywhere convergence of Htf to f

as t tends to 0, for f ∈ Lp(γ−1). The operator H ∗ is known to be bounded on Lp(γ−1) for

1 < p ≤ ∞, by standard Littlewood–Paley–Stein theory for symmetric diffusion semigroups

viii

[St1, C]. We shall prove that H ∗ is also of weak type 1 with respect to the measure γ−1.

Our proof follows the same lines of the proof of the corresponding result for the Ornstein–

Uhlenbeck semigroup [GMST1].

In Section 3.4 we state some results concerning a functional calculus for the operator A

on Lp(γ−1). We do not give proofs of these results, mainly because the proofs, though lengthy,

may be obtained from the corresponding results for the Ornstein–Uhlenbeck operator with

minor changes. Nevertheless, we believe that it is worth recording these results for future

reference. It is straightforward to show that the spectral resolution of A is

A =∞∑k=0

(k + n) Ek, (0.0.5)

where Ek is the orthogonal projection of L2(γ−1) onto the linear span of the functions of

the form γ1 p, and p is a Hermite polynomial of degree j in n variables. Given a bounded

sequence M : n, n + 1, . . . → C, we define the spectral multiplier operator associated to

the spectral multiplier M by

M(A )f =∞∑k=0

M(k + n)Ekf ∀f ∈ L2(γ−1). (0.0.6)

Clearly M(A ) is bounded on L2(γ−1); an interesting question is find necessary and/or suf-

ficient conditions on M so that M(A ) extends to a bounded operator on Lp(γ−1), for some

1 < p <∞, or of weak type 1 with respect to γ−1. Our results may be stated as follows:

(i) if M is the restriction to the L2-spectrum of A of a function M of Laplace transform

type, then M(A ) is of weak type (1, 1);

(ii) if 1 < p <∞ and u ∈ R, then

∣∣∣∣∣∣A iu∣∣∣∣∣∣Lp(γ−1)

eφ∗p|u| as u tends to ∞ :

here φ∗p = arcsin |2/p− 1|;

(iii) if 1 < p < ∞, p 6= 2, a is a positive number and M is the restriction of a bounded

holomorphic function on a + Sφ∗p and satisfies some mild conditions on the boundary,

then M(A ) is bounded on Lp(γ−1).

ix

(iv) if M is the restriction of a continuous function on [0,∞) and

supt>0

∣∣∣∣∣∣M(tA )∣∣∣∣∣∣Lp(γ−1)

<∞,

then M extends to a bounded holomorphic function on the sector Sφ∗p .

A few comments on these results are in order. By general semigroup theory [St1, C], if

1 < p <∞ and if M is the restriction to the L2-spectrum of A of a function M of Laplace

transform type, then A is bounded on Lp(γ−1). Thus, (i) above complements this result

by proving a limiting result for p = 1, which may be false for general symmetric diffusion

semigroups.

A result similar to (ii) holds for the Ornstein–Uhlenbeck operator [MMS, HMM]. Our

proof simplifies considerably the proof of the upper estimate given in [MMS]. The proof of the

lower estimate is very similar to that of the corresponding result for the Ornstein–Uhlenbeck

operator [HMM].

Recently, A. Carbonaro and O. Dragicevic [CD], improving previous result of Cowling [C]

and using a general multiplier result of Meda [Me], proved that all submarkovian semigroups

admit a functional calculus similar to that considered in (iii), but they need to assume that

a = 0, and require a slightly stronger condition on the boundary of Sφ∗p .

Finally (iv) is the analogue for A of a result for the Ornstein–Uhlenbeck operator

[HMM].

We will use the “variable constant convention”, and denote by C, possibly with sub-

or superscripts, a constant that may vary from place to place and may depend on any

factor quantified (implicitly or explicitly) before its occurrence, but not on factors quantified

afterwards.

Part I

Generalized Bergman spaces

and Hardy-type spaces

1

Chapter 1

Generalized Bergman spaces in Rn

1.1 Mean value properties for polyharmonic functions

For each x in Rn and r > 0 we denote by B(x, r) the Euclidean ball with centre x and

radius r and by B the family of all (open) balls in Rn. For each B in B we denote by cB

and rB the centre and the radius of B, respectively. For c > 0, we denote by cB the ball

with centre cB and radius c rB. We shall denote by B1 the unit ball centred at the origin

and by c(n) its Lebesgue measure. The Lebesgue measure of the measurable set E will be

denoted by |E|.It is well known that harmonic functions can be characterized by mean value properties.

In particular, a twice continuously differentiable function u is harmonic on a domain Ω ⊆ Rn

if and only if for every ball B(x, r) ⊆ Ω the average of u over B(x, r) is equal to u(x). A

similar result holds for polyharmonic functions. Recall that a function u is polyharmonic of

degree k (or k-harmonic) on Ω if u ∈ C 2k(Ω) and ∆ku = 0 on Ω (here ∆ denotes the Laplace

operator). Notice that, by elliptic regularity, if u is a distribution on Ω such that ∆ku = 0,

then u is in C 2k(Ω), hence it is k-harmonic in Ω.

In 1909 P. Pizzetti [Piz] proved, for n = 2, 3, a mean value property for polyharmonic

functions. He showed that the spherical mean of a polyharmonic function u on ∂B(x, r) may

be expressed as a linear combination of ∆ju(x), j = 1, . . . , k, with coefficients depending

on r. In 1936, M. Nicolesco [Nic] extended Pizzetti’s formula to all dimensions and to the

case of solid means. In particular, he proved the following.

Theorem 1.1.1. Assume that k is a positive integer and that Ω is a domain in Rn. If u is

3

4 CHAPTER 1. GENERALIZED BERGMAN SPACES IN RN

k-harmonic on Ω, then for every x ∈ Ω and r < d(x, ∂Ω)

B(x,r)

u =k−1∑j=0

d(j) ∆ju(x) r2j,

where d(0) = 1, d(j) = 1/(2j j! (n + 2) . . . (n + 2j)) for j ∈ 1, . . . , k − 1 andfflB(x,r)

u =

|B(x, r)|−1 ∫B(x,r)

u(y) dy.

By using this result, in 1966, J.H. Bramble and L.E. Payne [BP] proved the following char-

acterisation of k-harmonic functions.

Theorem 1.1.2. Assume that k is a positive integer and that Ω is a domain in Rn. The

following hold:

(i) if u is k-harmonic on Ω, then for every x ∈ Ω and r < d(x, ∂Ω), and for every choice

of k distinct values βj ∈ (0, 1), j = 1, . . . , k,

u(x) =k∑j=1

Cj

Bj

u, (1.1.1)

where Cj =∏m 6=j βm∏

m 6=j(βm−βj)and Bj = B(x,

√βjr);

(ii) if u ∈ C 2k(Ω) satisfies (1.1.1) for every x ∈ Ω, for every r < d(x, ∂Ω) and for every

choice of k distinct values βj ∈ (0, 1), j = 1, . . . , k, then u is k-harmonic in Ω.

1.2 Generalized Bergman spaces

Definition 1.2.1. Suppose that k is a positive integer, p is in [1,∞) and B is an open ball.

The generalised Bergman space bpk(B) is the space of all k-harmonic functions u which belong

to Lp(B), endowed with the Lp(B)-norm.

Observe that

bp1(B) ⊂ bp2(B) ⊂ · · · ⊂ Lp(B),

with proper inclusions. It is straightforward to check that for every integer k the Bergman

space bpk(B) is a closed subspace of Lp(B). Hence bpk(B) is a closed subspace of bpk+1(B).

1.2. GENERALIZED BERGMAN SPACES 5

In particular, when p = 2 we denote by Mk−1(B) the orthogonal complement of b2k−1(B) in

b2k(B). Thus,

b2k(B) = b2

k−1(B) ⊥Mk−1(B).

Working recursively, we obtain the following orthogonal decomposition of b2k(B)

b2k(B) = b2

1(B) ⊥M1 ⊥ · · · ⊥Mk−1(B). (1.2.1)

Clearly ∆k maps b2k+1(B) into the vector space of harmonic functions on B, but not neces-

sarily into b21(B), for functions in the range of ∆k may not belong to L2(B). Note that the

restriction of ∆k to Mk(B) is injective. Indeed, if u is in Mk(B) and ∆ku = 0, then u is in

b2k(B), whence u = 0, because b2

k(B) ∩Mk(B) = 0.

Next we examine the structure of b2k(B1) more closely (recall that B1 denotes the unit ball

in Rn centred at 0). Preliminarily, we prove some facts about polynomials in Rn. Denote by

Pm(Rn) the vector space of all homogeneous polynomials of degree m in Rn and by Hm(Rn)

the space of harmonic polynomials in Pm(Rn). A well known result [ABR, Proposition 5.5]

asserts that

Pm(Rn) = Hm(Rn)⊕ |x|2 Pm−2(Rn).

For the sake of brevity, we set

νkm := 2k k!k−1∏j=0

(n+ 2m+ 2j

). (1.2.2)

Lemma 1.2.2. The operator ∆k is an isomorphism between |x|2kHm(Rn) and Hm(Rn), with

inverse

∆−kh(x) =1

νkm|x|2k h(x) ∀h ∈Hm(Rn).

Proof. We observe that if h is in Hm, then ∆k(|·|2k h) is a multiple of h, hence harmonic.

Indeed, by Leibnitz’s formula and Euler’s formula,

∆(|·|2k h)(x) = (∆ |x|2k)h(x) + 2∇ |x|2k · ∇h(x) + |x|2k ∆h(x)

= 2k(n+ 2(k − 1)) |x|2(k−1) h(x) + 4k |x|2(k−1) x · ∇h(x)

= 2k(n+ 2m+ 2(k − 1)) |x|2(k−1) h(x).

By arguing recursively, we see that

∆k(|·|2k h) = νkm h. (1.2.3)


Therefore ∆k is a bijection between |x|2kHm(Rn) and Hm(Rn), and its inverse ∆−k is given

by

∆−kh(x) =1

νkm|x|2k h(x) ∀h ∈Hm(Rn),

as required.

Lemma 1.2.3. Suppose that p is a homogeneous polynomial of degree j in b2k(B1). Then

there exist unique homogeneous harmonic polynomials pj−2m ∈Hj−2m(Rn), m = 0, . . . , k−1,

such that

p(x) = pj(x) + |x|2 pj−2(x) + · · ·+ |x|2(k−1) pj−2(k−1)(x). (1.2.4)

Proof. Set J = [j/2]. By a well known result [ABR, Theorem 5.7], there exist unique

homogeneous harmonic polynomials pj−2mJm=0, with pj−2m of degree j − 2m, such that

p(x) = pj(x) + |x|2 pj−2(x) + · · ·+ |x|2J pj−2J(x).

We now impose the condition ∆kp = 0. By (1.2.3), if k > m, then

∆k(|·|2m pj−2m) = ∆k−m∆m(|·|2m pj−2m) = νmj−2m ∆k−mpj−2m = 0,

because pj−2m is harmonic. If, instead, k ≤ m, then, by arguing as in the proof of

Lemma 1.2.2, we see that

∆k(|·|2m pj−2m) = 2km!

(m− k)!

k∏i=1

(n+ 2j − 2m− 2i) |·|2(m−k) pj−2m.

Altogether, we see that ∆kp is a linear combination with nonvanishing coefficients of the

polynomials |x|2(m−k) pj−2m, where m = k − 1, . . . , J . Observe that ∆kp is homogeneous of

degree j−2k. Therefore, by the uniqueness in the decomposition of homogeneous polynomials

[ABR, Theorem 5.7], ∆kp = 0 if and only if pj−2m = 0 for all m = k − 1, . . . , J , as required.

Remark 1.2.4. For later purposes, it is desirable to decompose every homogeneous polynomial

of degree j in b2k(B1) according to the orthogonal decomposition (1.2.1). Formula (1.2.4)

expresses a homogeneous polynomial p of degree j in b2k(B1) as the sum of the homogeneous

polynomials pj, |·|2pj−2, . . . , |·|2(k−1)pj−2(k−1), which belong to b21(B1), b2

2(B1), . . . , b2k−1(B1),

respectively. These are pairwise orthogonal in L2(B1), because, for m 6= l,∫B1

|x|2m+2l pj−2m(x) pj−2l(x) dx =

∫ 1

0

sn+2j−1 ds

∫Sn−1

pj−2m(ω) pj−2l(ω) dω,


and spherical harmonics of different degrees are orthogonal on the unit sphere. Here, pj−2l(ω)

denotes the complex conjugate of pj−2l(ω). However, this is not the desired orthogonal

decomposition, because, for instance, pj−2m, which belongs to b21(B1), and |x|2j pj−2m are

obviously not orthogonal in L2(B1).

We now explain how to perform the desired decomposition. It may be worth dealing first

with the case k = 2. Given f in b2k(B1), denote by Π0f the orthogonal projection of f onto

b21(B1), and, for j = 1, . . . , k−1, denote by Πjf the orthogonal projection of f onto Mj(B1).

Lemma 1.2.5. Let p be a homogeneous polynomial of degree j in b22(B1) and let pj and pj−2

be as in (1.2.4). Then

Π0p = pj +n+ 2j − 4

n+ 2j − 2pj−2 and Π1p =

(|·|2 − n+ 2j − 4

n+ 2j − 2

)pj−2.

Proof. It is clear that p = Π0p+ Π1p and that pj + (n+ 2j−4)/(n+ 2j−2)pj−2 is harmonic.

Denote by p1j−2 the polynomial defined by

p1j−2(x) =

(|x|2 − n+ 2j − 4

n+ 2j − 2

)pj−2(x) (1.2.5)

By Lemma 1.2.2, p1j−2 is in b2

2(B1). Thus, it remains to show that p1j−2 is orthogonal to all

harmonic polynomials, for their restrictions to B1 are dense in b21(B1). Suppose that qk is a

homogeneous harmonic polynomial of degree k. Then∫B1

qk(x)(|x|2 − n+ 2j − 4

n+ 2j − 2

)pj−2(x) dx

=

∫ 1

0

sn+k+j−3(s2 − n+ 2j − 4

n+ 2j − 2

)ds

∫Sn−1

qk(ω) pj−2(ω) dω.

If j− 2 6= k, then the inner integral vanishes, for spherical harmonics of different degrees are

orthogonal. If j − 2 = k, then the outer integral vanishes, as a straighforward calculation

shows, and the required result follows.

We aim at extending the decomposition above to b2k(B1). The extension hinges on the

following technical lemma.

Lemma 1.2.6. There exists a unique sequence of polynomials αkj : j, k = 0, 1, 2, . . . on Rwith the following properties:

(i) αkj is a monic even polynomial of degree 2k for every j in N;


(ii) for every k ≥ 1 the following orthogonality relations hold∫ 1

0

αkj (s)αlj(s) s

n+2j−1 ds = 0 l = 0, . . . , k − 1, j = 0, 1, 2, . . . . (1.2.6)

Furthermore ∫ 1

0

αkj (s) sn+2j−1+2k ds 6= 0 j = 0, 1, 2, . . . . (1.2.7)

For k ≥ 1, the polynomial αkj is given by

αkj (s) = s2k +k−1∑i=0

Cki (j, n) s2i, (1.2.8)

where

Cki (j, n) = (−1)k−i

(k

i

) ∏k−1l=i (n+ 2j + 2l)∏2k−1l=k+i(n+ 2j + 2l)

. (1.2.9)

Hereafter, to simplify notation, we shall often omit the dependence of the coefficients Cki (j, n)

on j and n, and write simply Cki . We shall prove that the polynomials αkj are given by (1.2.8)

and (1.2.9) by brute force and, although conceptually very simple, the proof requires long and

tedious calculation. The precise form of the coefficients will be used only later to estimate

the generalized Bergman projections. Therefore we have chosen to postpone this part of the

proof of the lemma to the Appendix to this chapter.

Proof. (of the existence and uniqueness of αkj ) For the duration of this proof, for each j in Nwe shall denote by (·, ·)j the inner product in L2([0, 1], sn+2j−1 ds). Observe preliminarly

that, for each k ≥ 1, (1.2.6) is equivalent to the following:∫ 1

0

αkj (s) sn+2j+2m−1 ds = 0 m = 0, 1, . . . , k − 1 j = 0, 1, 2, . . . . (1.2.10)

Furthermore, if αkj exists, then, by (i), it must be of the form:

αkj (s) = s2k +k−1∑i=0

Cki (j, n) s2i,

where the coefficients Cki (j, n) must be determined.

We argue by induction on k. Suppose first that k = 1. Then a straightforward calculation

shows that for every nonnegative integer j the system (1.2.10) has a unique solution, given by

α1j (s) = s2 − n+ 2j

n+ 2j + 2,


(see also Lemma 1.2.5). Furthermore, (α1j , s

2)j 6= 0.

Now suppose that for each ` ≤ k − 1, there exists a unique sequence α`j : j ∈ Nof polynomials with the required properties (in particular (α`j, s

2`)j 6= 0). Since (1.2.6) is

equivalent to (1.2.10), (1.2.10), with ` in place of k, implies that (α`j, s2m)j = 0 for each

m ∈ 0, . . . , `− 1. Therefore,

(αkj , α`j)j =

∫ 1

0

(s2k +

k−1∑i=`

Cki s

2i)α`j(s) s

n+2j−1 ds = 0 ∀` = 0, . . . , k − 1. (1.2.11)

Observe that (1.2.11) is an upper-triangular nonhomogeneous system with unknowns Ck0 , . . . ,

Ckk−1. Hence it has a unique solution if and only if the diagonal entries of the associated

matrix do not vanish. These are∫ 1

0

α`j(s) sn+2j−1+2` ds ` = 0, . . . , k − 1,

which indeed do not vanish by the inductive hypothesis. This concludes the proof of the

inductive step, and of the first part of the lemma.

With a slight abuse of notation, we also denote by αkj the polynomial on Rn, defined by

αkj (x) = |x|2k + Ckk−1 |x|

2(k−1) + · · ·+ Ck0 , (1.2.12)

where the coefficients Cki k−1

i=0 are as in Lemma 1.2.6. Denote by P the linear space of all

polynomials in Rn. For each j in N and each k ≥ 1, define the map E kj : Hj →P by

E kj p = αkj p ∀p ∈Hj. (1.2.13)

It will be convenient to denote the identity map by E 0j : Hj →Hj. For each ` ≥ 1, set

Q` := span

Ran(Em−1j ) : j ∈ N, m = 1, . . . , `

.

For j in N, let pj,1, . . . , pj,dj be any orthonormal basis of Hj.

Lemma 1.2.7. Suppose that k ≥ 1. The following hold:

(i) Ran(E k−1j

)is contained in b2

k(B1) and

Ran(E k−1j

)⊥ Ran

(Em−1j

)∀m ∈ 1, . . . , k − 1;


(ii) Qk is dense in b2k(B1);

(iii) spanRan(E k−1j : j ∈ N

) is dense in Mk−1(B1).

(iv) the sequence

E k−1j pj,1, . . . ,E

k−1j pj,dj j = 0, 1, 2, . . .

is an orthogonal basis of Mk−1(B1). Hence

E hj pj,1, . . . ,E

hj pj,dj h = 0, . . . , k − 1, j = 0, 1, 2, . . .

is an orthogonal basis of b2k(B1).

Proof. First we prove (i). Observe that

E k−1j p = |·|2(k−1) p+ Ck−1

k−2 |·|2(k−2) p+ · · ·+ Ck−1

0 p,

whence

∆k(E k−1j p

)= ∆∆k−1

(|·|2(k−1) p

)+ Ck−1

k−2 ∆2∆k−2(|·|2(k−2) p

)+ · · ·+ Ck−1

0 ∆k−1∆p.

For ` ≥ 1, the polynomial ∆`(|·|2` p

)is harmonic by Lemma 1.2.2. Therefore all the sum-

mands on the right hand side vanish, whence E k−1j p is in b2

k(B1).

Now, suppose that p and q are homogeneous harmonic polynomials of degree j and h,

respectively. Then

(E k−1j p,Em−1

h q) =

∫B1

αk−1j (x) p(x) αm−1

h (x) q(x) dx

=

∫ 1

0

αk−1j (s)αm−1

h (s) sn+j+h−1 ds

∫Sn−1

p(ω) q(ω) dω.

(1.2.14)

The inner integral vanishes if j 6= h (spherical harmonics of different degrees are orthogonal

on Sn−1), and the outer integral vanishes if j = h, by (1.2.6). This concludes the proof of (i).

Next we prove (ii). It is well known (see [ABR, Corollary 5.34]) that the span of all

harmonic homogeneous polynomials is dense in b21(B1), hence the result holds for k = 1.

Henceforth we assume k ≥ 2. We claim that the space of all u in b2k(B1) that extend to a

bounded k-harmonic function in a neighbourhood of B1 is dense in b2k(B1). Indeed, suppose

that u is in b2k(B1). Then for every r ∈ (0, 1), the r-dilate ur of u, defined by

ur(x) = u(rx) ∀x ∈ (1/r)B1,


is in b2k((1/r)B1), hence it is k-harmonic and bounded in a neighbourhood of B1. The claim

then follows from the fact that ur tends to u in b2k(B1).

Thus, to prove that Qk is dense in b2k(B1) it suffices to prove that every function u in

bk2(B1) that extends to a k-harmonic function in a neighbourhood of B1 may be approximated

with arbitrary degree of precision by polynomials in Qk.

We argue by induction on k. As we have already said, the property holds for k = 1. Now

suppose that Q` is dense in b2`(B1) for ` = 1, . . . , k − 1, and let u be a function in b2

k(B1)

that extends to a k-harmonic function in RB1 for some R > 1. Then ∆k−1u is harmonic in

RB1. Hence there exist homogeneous harmonic polynomials pj : j ∈ N such that

∆k−1u =∞∑j=0

pj, (1.2.15)

where pj is a homogeneous harmonic polynomial of degree j. It is known [ABR, Corol-

lary 5.34] that the series (1.2.15) is absolutely and uniformly convergent in every compact

subset of RB1. In particular, for every r in (1, R)

limj→∞

supx∈rB1

|pj(x)| = 0.

Therefore, for every r′ in (1, r)

supx∈r′B1

|pj(x)| = sup0≤s≤r′

supx′∈Sn−1

sj |pj(x′)|

= sup0≤s≤r′

supx′∈Sn−1

(sr

)jrj |pj(x′)|

≤ C(r′r

)j∀j ∈ N.

Define U by

U =∞∑j=0

∆1−kpj, (1.2.16)

where

∆1−kpj(x) =(νk−1j

)−1 |x|2(k−1) pj(x)

and νk−1j is as in Lemma 1.2.2. Note that νk−1

j jk as j tends to infinity. Hence the series

(1.2.16) is absolutely and uniformly convergent in every compact subset of RB1. Moreover

the estimates above imply that ∆k−1U =∑

j pj, which is also equal to ∆k−1u. Therefore

∆k−1(U −u) = 0 in RB1, i.e., U −u is a (k−1)-harmonic function in RB1. By the induction


hypothesis, U − u may be approximated to any degree of precision by (k − 1)-harmonic

polynomials. Thus, to prove that u may be well approximated by k-harmonic polynomials,

it suffices to show that U does. This is straightforward, for the series (1.2.16) converges

uniformly in a neighbourhood of B1, hence in L2(B1). This concludes the proof of the

inductive step, and of (ii).

Part (iii) is a direct consequence of (i) and (ii).

To prove (iv), observe that, by (iii), the span of

E k−1j pj,1, . . . ,E

k−1j pj,dj j = 0, 1, 2, . . . (1.2.17)

is dense in Mk−1(B1). Furthermore

(E k−1j pj,`,E

k−1h pj,m) =

∫ 1

0

αk−1j (s)2 sn+2j−1 ds

∫Sn−1

pj,`(ω) pj,m(ω) dω

= 0

(1.2.18)

because pj,` and pj,m are orthogonal in L2(B1), whence so are their restrictions to Sn−1.

Therefore (1.2.17) is an orthogonal basis of Mk−1(B1). The last statement of (iv) follows

from this and the orthogonal decomposition (1.2.1).

This concludes the proof of the lemma.

1.3 The reproducing kernel of generalized Bergman

spaces

Let B be an open ball in Rn and suppose that p is in [1,∞). For each x in B and each

multiindex γ, denote by Λγx the evaluation functional at x on bpk(B), defined by

Λγxu = Dγu(x) ∀u ∈ bpk(B).

A noteworthy consequence of the mean value property for k-harmonic functions (1.1.1) is

that Λγx is continuous on bpk(B), as shown in the next proposition.

Proposition 1.3.1. Suppose that p is in [1,∞), that x is in B and that γ is a multiindex.

Then there exists a constant C, depending only on n, k and γ, such that∣∣∣∣∣∣Λγx

∣∣∣∣∣∣bpk≤ C

d(x, ∂B)|γ|+n/p. (1.3.1)

1.3. THE REPRODUCING KERNEL OF GENERALIZED BERGMAN SPACES 13

Proof. The proof is by induction on |γ|. First we consider the case where |γ| = 0. Let

r < d(x, ∂B). Set βj = 1− 2−j. It is straighforward to check that∣∣∣∏m 6=j

βmβm − βj

∣∣∣ ≤ 2k(k−1).

Thus, |Cj| ≤ 2k(k−1), for j = 1, . . . , k, where Cj is defined in (1.1.1). Hence, by the mean

value property (1.1.1),

|u(x)| ≤ Ck∑j=1

1∣∣B(x,√βjr)

∣∣∫B(x,√βjr)

|u(y)| dy

≤ Ck∑j=1

‖u‖p∣∣B(x,√βjr)

∣∣1/p≤ C r−n/p ‖u‖p.

We have used Holder’s inequality in the second inequality above. Then (1.3.1) (with γ = 0)

follows by letting r tend to d(x, ∂B).

Next, suppose that (1.3.1) holds for all multiindices γ such that |γ| ≤ m, and suppose

that |γ′| = m + 1. Then there exists i ∈ 1, . . . , n such that Dγ′ = ∂iDγ, where |γ| = m.

Choose r < d(x, ∂B)/2. Observe that, by the mean value property of k-harmonic function

applied to ∂iDγu and by translation invariance of the Lebesgue measure,

∂iDγu(x) =

k∑j=1

Cj∣∣B(0,

√βjr)

∣∣−1∫B(0,√βjr)

∂iDγu(x+ y) dy.

By the divergence theorem and the inductive hypothesis

|∂iDγu(x)| ≤ C

rn

k∑j=1

∫∂B(0,

√βir)

|Dγu(x+ ω)| dω

≤ C

rn

k∑j=1

‖u‖p[

inf|ω|=r

d(x+ ω, ∂B)]−|γ|−n/p ∫

∂B(0,√βir)

dω

≤ C r−|γ|−1−n/p ‖u‖p,

from which the required estimate for∣∣∣∣∣∣Λγ′

x

∣∣∣∣∣∣bpk

follows.


Definition 1.3.2. Denote by RkB(x, ·) the unique function in b2

k(B) that represents the

continuous linear functional Λ0x. Then

u(x) =

∫B

RkB(x, y)u(y) dy ∀u ∈ b2

k(B). (1.3.2)

The function RkB on B×B will be called the (generalized) Bergman kernel or the reproducing

kernel of b2k(B).

In the following proposition we collect some elementary properties of RkB.

Proposition 1.3.3. The following hold:

(i) RkB is real-valued;

(ii) if un is an orthonormal basis of b2k(B), then

RkB(x, y) =

∞∑j=1

uj(x)uj(y) ∀x, y ∈ B;

(iii) RkB(x, y) = Rk

B(y, x) for all x and y in B;

(iv) ‖RkB(x, ·)‖b2k = Rk

B(x, x)1/2

for all x in B.

Proof. These are standard properties of reproducing kernels. The proof is almost identical

to the proof of [ABR, Proposition 8.4], and is omitted.

Recall the orthogonal decomposition

b2k(B) = b2

1(B) ⊥M1(B) ⊥ · · · ⊥Mk−1(B). (1.3.3)

Each of the subspaces of b2k(B) that appear on the right hand side is closed in b2

k(B). There-

fore the restriction of Λ0x to Mj(B) is a continuous linear functional on Mj(B), so that, by

the Riesz representation theorem, there exists a unique function RMj

B (x, ·) in Mj(B) that

represents Λ0x∣∣Mj

.

Proposition 1.3.4. Suppose that B is an open ball in Rn. The following hold:

(i) if B has radius r, then

RkB(x, y) = r−nRk

B1

(x− cBr

,y − cBr

)∀(x, y) ∈ B ×B; (1.3.4)


(ii) RkB = R1

B +RM1B + · · ·+R

Mk−1

B .

Proof. First we prove (i). Suppose that u is in b2k(B). Then the function

τcBur(x) := u(rx+ cB) ∀x ∈ B1

is in b2k(B1). Since Rk

B1is the reproducing kernel for b2

k(B1),

u(rx+ cB) =

∫B1

τcBur(y)Rk

B1(x, y) dy

= r−n∫B

u(y)RkB1

(x, (y − cB)/r) dy ∀x ∈ B1,

which may be rewritten in the form

u(x) = r−n∫B

u(y)RkB1

((x− cB)/r, (y − cB)/r) dy ∀x ∈ B,

and (i) is proved.

Next we prove (ii). Suppose that u is in b2k(B). Then there exist u0 in b2

1(B), uh in

Mh(B), h = 1, . . . , k − 1, such that u = u0 + u1 + · · ·+ uk−1. Then

u(x) = u0(x) + u1(x) + · · ·+ uk−1(x)

=

∫B

R1B(x, y)u0(y) dy +

k−1∑h=1

∫B

RMhB (x, y)uh(y) dy

Since the decomposition (1.3.3) is orthogonal, the right hand side is equal to∫B

R1B(x, y)u(y) dy +

k−1∑h=1

∫B

RMhB (x, y)u(y) dy.

Therefore R1B(x, ·) + RM1

B (x, ·) + · · · + RMk−1

B (x, ·) represents Λ0x on b2

k(B). By uniqueness,

this must be RkB(x, ·), as required.

Our aim is to establish an explicit formula for RkB. By Proposition 1.3.4 (i) it suffices to

determine RkB1

. It is known [ABR, Theorem 8.9] that

R1B1

(x, y) =1

n c(n)

∞∑j=0

(n+ 2j) Zj(x, y), (1.3.5)

where

Zj(x, y) := |x|j |y|j Zj( x|x|,y

|y|

)∀x, y ∈ Rn


and Zj denotes the jth zonal harmonic on Sn−1. Clearly Zj(x, ·) is in Hj. For any j ∈ N,

define σhj by the rule (σhj )−1 =

∫ 1

0

αhj (s)2 sn+2j−1 ds,

where αhj is as in (1.2.8). Observe that

(σhj)−1

=

∫ 1

0

αhj (s) sn+2j+2h−1 ds,

for αhj (s) is orthogonal to s2m for each m in 0, . . . , h−1, by (1.2.10). Then formula (1.3.15),

with k replaced by h and m = h, gives

σhj =n+ 2j + 4h

22h (h!)2

2h−1∏l=h

(n+ 2j + 2l)2. (1.3.6)

Proposition 1.3.5. The reproducing kernel of Mh(B1) is given by

RMhB1

(x, y) =1

n c(n)

∞∑j=0

σhj αhj (x) Zj(x, y)αhj (y) ∀x, y ∈ B1. (1.3.7)

Proof. For each j in N let pj,1, . . . , pj,dj be an orthonormal basis of Hj.

We claim that the function Rhj , given by

Rhj (x, y) =

1

n c(n)σhj α

hj (x) Zj(x, y)αhj (y) ∀x, y ∈ B1,

is the reproducing kernel of E hj Hj. Clearly, it suffices to show that(

E hj pj,i, R

hj (x, ·)

)= E h

j pj,i(x) i = 1, . . . , dj.

To prove this, notice that

(E hj pj,i, R

hj (x, ·)

)=σhj α

hj (x)

n c(n)

∫B1

αhj (y) pj,i(y) Zj(x, y)αhj (y) dy

= σhj αhj (x) |x|j

∫ 1

0

αhj (s)2 sn+2j−1 ds

∫Sn−1

Zj(x/ |x| , ω) pj,i(ω) dσ(ω)

= αhj (x) |x|j pj,i(x/ |x|)

= E hj pj,i(x) :

we have used the fact that Zj is the reproducing kernel of spherical harmonics of degree j

in the third equality above.


Observe that if j 6= `, then(Rh` (x, ·), Rh

j (x, ·))

=1

n2 c(n)2σhj σ

h` α

hj (x)αh` (x)

(E h` Z`(x, ·),E h

j Zj(x, ·))

= 0,

because Z`(x, ·) is in H`, Zj(x, ·) is in Hj, and the ranges of E h` and E h

j are orthogonal by

(1.2.14).

Next we show that for every x in B1 the series∑

j Rj(x, ·) is convergent in L2(B1).

Indeed, observe that there exists a constant C such that

h−1∑i=0

∣∣Chi (j, n)

∣∣ ≤ C ∀j ∈ N.

This may be easily deduced from (1.2.9), with h in place of k, and the fact that∏h−1l=i (n+ 2j + 2l)∏2h−1l=h+i(n+ 2j + 2l)

≤ 1.

Therefore ∣∣αhj (x)∣∣ ≤ C ∀x ∈ B1 ∀j ∈ N,

and ∥∥Rhj (x, ·)

∥∥L2(B1)

≤ C σhj ‖Zj(x, ·)‖L∞(B1) ‖αhj ‖L2(B1)

≤ C (σhj )1/2 |x|j dim(Hj).

We have used the well known estimate∣∣Zj(x, ·)∣∣ ≤ dim(Hj) and the fact that (σhj )−1/2 =

‖αhj ‖L2(B1) (see the definition of σhj ) in the second inequality. Now, from (1.3.6) we deduce

that (σhj )1/2 jh+1/2 as j tends to infinity. (1.3.8)

Furthermore

dim(Hj) jn−2 as j tends to infinity.

Altogether, we have proved that there exists a constant C, independent of j, such that∥∥Rhj (x, ·)

∥∥L2(B1)

≤ C jn+h−3/2 |x|j .

This implies that the series∑

j Rhj (x, ·) is convergent in L2(B1). Clearly

∑j R

hj (x, ·) repro-

duces all polynomials in span(⋃∞

j=0 E hj Hj

), which is dense in Mh(B1) by Lemma (1.2.7) (iii).

A density argument then shows that∑

j Rhj (x, ·) is a reproducing kernel of Mh(B1). Hence

it must be RMhB1

(x, ·), by uniqueness, as required.


1.3.1 Appendix

In this appendix, we present the proof of second part of Lemma 1.2.6. It would be highly

desirable to find a simpler proof thereof.

Proof. (of the second part of Lemma 1.2.6) We prove that the polynomial (1.2.8) with

coefficients (1.2.9) satifies condition (1.2.10). For notational convenience, set

β(0)j,k (s) := αkj (s) and β

(i)j,k(s) =

(β(i−1)j,k )′(s)

2si = 1, 2, . . . .

We integrate repeatedly by parts in (1.2.10), and obtain, for every m ∈ N,

∫ 1

0

αkj (s) sn+2j+2m−1 ds =

[β(0)j,k (s) sn+2j+2m

n+ 2j + 2m

]1

0− 2

n+ 2j + 2m

∫ 1

0

β(1)j,k (s) sn+2j+2m+1 ds

=β

(0)j,k (1)

n+ 2j + 2m− 2

n+ 2j + 2m

∫ 1

0

β(1)j,k (s) sn+2j+2m+1 ds

= . . .

=k∑i=0

(−1)i 2i β(i)j,k(1)∏i

l=0(n+ 2j + 2(m+ l)).

We claim that

β(0)j,k (1) =

2k k!∏2k−1l=k (n+ 2j + 2l)

. (1.3.9)

For the rest of this proof, we set ηl := n+ 2j + 2l. Clearly, β(0)j,k (1) = 1 +

∑k−1i=0 C

ki . To prove

the claim, we first show that

β0j,k(1)

2N k . . . (k −N + 1)=

k−N−1∑i=0

(−1)k−N−i(k −Ni

) ∏k−1l=i+N ηl∏2k−1l=k+i ηl

+1∏2k−1

l=2k−N ηl(1.3.10)

for each N ∈ 1, . . . , k−1. We argue by finite induction on N . Consider 1+∑

iCki . It is an

elementary fact that(ki

), which is one of the factors that appear in formula (1.2.9) for Ck

i , is

equal to(k−1i

)if i = 0, and decomposes as

(k−1i

)+(k−1i−1

)if i ≥ 1. Observe that

(k−1i

)is also

one of the two binomial coefficients which give the analogous decomposition of(ki+1

), that

appears in Cki+1. Moreover, Ck

i and Cki+1 share

∏k−1l=i+1 ηl/

∏2k−1l=k+i+1 ηl. It is then convenient


to sum up by collecting like terms, yielding

k−1∑i=0

Cki + 1 =

k−2∑i=0

(−1)k−i−1

(k − 1

i

) ∏k−1l=i+1 ηl∏2k−1

l=k+i+1 ηl

[1− n+ 2j + 2i

n+ 2j + 2(k + i)

]+ 1− n+ 2j + 2(k − 1)

n+ 2j + 2(2k − 1)

=k−2∑i=0

(−1)k−i−1

(k − 1

i

)2k∏k−1

l=i+1 ηl∏2k−1l=k+i ηl

+2k

n+ 2j + 2(2k − 1),

which gives (1.3.10) for N = 1.

The inductive step may be completed by similar calculations. For, consider the right hand

side of (1.3.10). We write(k−N−1

0

)instead of

(k−N

0

)and

(k−N−1

i

)+(k−N−1i−1

)instead of

(k−Ni

).

Now, for each i = 0, . . . , k−N−2, the expression(k−N−1

i

)appears two times, the first multi-

plied by (−1)k−N−i∏k−1

l=i+N ηl/∏2k−1

i=k+i ηl, the second by (−1)k−N−i−1∏k−1

l=i+N+1 ηl/∏2k−1

i=k+i+1 ηl.

The sum of these two terms equals

(−1)k−N−i−1

(k −N − 1

i

) ∏k−1l=i+N+1 ηl∏2k−1l=k+i+1 ηl

[1− n+ 2j + 2(i+N)

n+ 2j + 2(k + i)

],

which is

(−1)k−N−i−1

(k −N − 1

i

)2(k −N)

∏k−1l=i+N+1 ηl∏2k−1

l=k+i ηl.

Similarly, we sum up −(n+2j+2(k−1))/∏2k−1

l=2k−N−1 ηl, which comes from the decomposition

of the term corresponding to i = k −N − 1, and 1/∏2k−1

l=2k−N ηl, obtaining

1∏2k−1l=2k−N ηl

[1− n+ 2j + 2(k − 1)

n+ 2j + 2(2k −N − 1)

]=

2(k −N)∏2k−1l=2k−N−1 ηl

.

It follows that

1 +∑k−1

i=0 Cki

2N k . . . (k −N + 1)

=k−N−2∑i=0

(−1)k−N−i−1

(k −N − 1

i

)2(k −N)

∏k−1l=i+N+1 ηl∏2k−1

l=k+i ηl+

2(k −N)∏2k−1l=2k−N−1 ηl

,

which is (1.3.10) for N + 1.


Now, the claim follows easily. Indeed, by (1.3.10) with N = k − 1,

1 +k−1∑i=0

Cki = 2k−1 k!

(− n+ 2j + 2(k − 1)∏2k−1

l=k ηl+

1∏2k−1l=k+1 ηl

)=

2k−1 k!∏2k−1l=k+1 ηl

(1− n+ 2j + 2(k − 1)

n+ 2j + 2k

)=

2k k!∏2k−1l=k ηl

,

and (1.3.9) is proved.

Similar arguments lead to

β(i)j,k(1) =

2k−i k!(ki

)∏2k−1l=k+i ηl

∀i ∈ 1, . . . , k − 1. (1.3.11)

For the sake of completeness we give a sketch of the proof of this fact as well.

Similarly as before, we need to prove the auxiliary relation

β(i)j,k(1)

k . . . (k − i+ 1)= 2N k . . . (k −N + 1)

×

k−N−1∑h=i

(−1)k−N−h(k − i−Nh− i

)∏k−1l=h+N ηl∏2k−1l=k+h ηl

+1∏2k−1

l=2k−N ηl

(1.3.12)

for each N ∈ 1, . . . , k − i− 1.Once again, we proceed by finite induction on N . To prove (1.3.12) for N = 1, we observe

that clearly

β(i)j,k(1) = k . . . (k − i+ 1) +

k−1∑h=i

h . . . (h− i+ 1)Ckh .

An easy computation shows that, for each h ∈ i, . . . , k−1, the coefficient h . . . (h− i+ 1),

multiplied by(kh

)(which appears in Ck

h), gives k . . . (k − i+ 1)(k−ih−i

). It follows that

β(i)j,k(1)

k . . . (k − i+ 1)= 1 +

k−1∑h=i

(−1)k−h(k − ih− i

) ∏k−1l=h ηl∏2k−1l=k+h ηl

.

We may now write(k−i−1

0

)instead of

(k−i

0

), decompose

(k−ih−i

)as(k−i−1h−i

)+(k−i−1h−i−1

)for h in

i+ 1, . . . , k − 1, and sum up by collecting like terms similarly as before. We obtain

β(i)j,k(1)

k . . . (k − i+ 1)= 2k

k−2∑h=i

(−1)k−h−1

(k − i− 1

h− i

) ∏k−1l=h+1 ηl∏2k−1l=k+h ηl

+1

n+ 2j + 2(2k − 1)

,


which is (1.3.12) for N = 1.

The inductive step can be proved by a similar argument. We omit the details.

Now, by (1.3.12) with N = k − i− 1,

β(i)j,k(1)

k . . . (k − i+ 1)= 2k−i−1 k . . . (i+ 2)

−n+ 2j + 2(k − 1)∏2k−1

l=k+i ηl+

1∏2k−1l=k+i+1 ηl

=2k−i−1 k . . . (i+ 2)∏2k−1

l=k+i+1 ηl

(1− n+ 2j + 2(k − 1)

n+ 2j + 2(k + i)

)=

2k−i k . . . (i+ 1)∏2k−1l=k+i ηl

.

To conclude the proof of (1.3.11), it suffices to observe that

k . . . (k − i+ 1) k . . . (i+ 1) = k!k!

(k − i)! i!= k!

(k

i

).

Finally, it is easy to see that β(k)j,k (1) = k!. This, together with (1.3.9) and (1.3.11), gives∫ 1

0

αkj (s) sn+2j+2m−1 ds

=2k k!

n+ 2j + 2m

(k0

)∏2k−1l=k ηl

+k−1∑i=1

(−1)i(ki

)∏2k−1l=k+i ηl

∏il=1(n+ 2j + 2(m+ l))

+(−1)k

(kk

)∏kl=1(n+ 2j + 2(m+ l))

.

(1.3.13)

We now use a procedure similar to that used to calculate (1.3.9) and (1.3.11) to reduce

further this expression. We claim that∫ 1

0

αkj (s) sn+2j+2m−1 ds

=2k k! 2N (k −M − 1) . . . (k −M −N)

n+ 2j + 2m

×k−N∑i=0

(−1)i+N(k−Ni

)∏2k−1l=k+i ηl

∏i+Nl=1 (n+ 2j + 2(m+ l))

(1.3.14)

for each N ∈ 1, . . . , k.We first prove the claim for N = 1. Once more, for each i ∈ 1, . . . , k − 1, we decompose

the term corresponding to i in (1.3.13) into the sum of two terms, by writing(k−1i

)+(k−1i−1

)


instead of(ki

). Moreover, we write

(k−1

0

)instead of

(k0

)and

(k−1k−1

)instead of

(kk

). We then

sum up(k−1

0

)/∏2k−1

l=k ηl and the second term obtained by the decomposition of the term

associated to i = 1, and obtain

−(k−1

0

)∏2k−1l=k+1 ηl

(1

n+ 2j + 2k− 1

n+ 2j + 2(m+ 1)

),

which equals−(k−1

0

)2(k −m− 1)∏2k−1

l=k ηl (n+ 2j + 2(m+ 1)).

Similarly, for any i ∈ 1, . . . , k − 2, we sum up the first term from the decomposition

of the term corresponding to i and the second term from the decomposition of the one

corresponding to i+ 1, yielding

(−1)i+1(k−1i

)∏2k−1l=k+i+1 ηl

∏il=1(n+ 2j + 2(m+ l))

(1

n+ 2j + 2(m+ i+ 1)− 1

n+ 2j + 2(k + i)

),

which is equal to(−1)i+1

(k−1i

)2(k −m− 1)∏2k−1

l=k+i ηl∏i+1

l=1(n+ 2j + 2(m+ l)).

Finally, the binomial coefficient(k−1k−1

)appears in one of the two terms in which the term

corresponding to i = k − 1 is decomposed. We sum up such term with the last term in

(1.3.13), and we obtain

(−1)k(k−1k−1

)∏k−1l=1 (n+ 2j + 2(m+ l))

(1

n+ 2j + 2(m+ k)− 1

n+ 2j + 2(2k − 1)

),

which is(−1)k

(k−1k−1

)2(k −m− 1)

(n+ 2j + 2(2k − 1))∏k

l=1(n+ 2j + 2(m+ l)).

Eventually, we obtain∫ 1

0

αkj (s) sn+2j+2m−1 ds = 2(k −m− 1)

k−1∑i=0

(−1)i+1(k−1i

)∏2k−1l=k+i ηl

∏i+1l=1(n+ 2j + 2(m+ l))

,

which is (1.3.14) for N = 1.

The proof of the fact that, if (1.3.14) holds for a certain N , then it also holds for N + 1, is

similar and is omitted.

1.4. ESTIMATES FOR GENERALIZED BERGMAN KERNELS 23

In particular, (1.3.14) with N = k gives∫ 1

0

αkj (s) sn+2j+2m−1 ds =

22k k!∏k−1

l=0 (m− l)∏2k−1l=k ηl

∏kl=0(n+ 2j + 2(m+ l))

, (1.3.15)

which clearly vanishes if m ∈ 0, . . . , k− 1. This concludes the proof of (1.2.10) and of the

lemma.

1.4 Estimates for generalized Bergman kernels

In this section we prove pointwise estimates for the reproducing kernel RkB of b2

k(B) that

generalise those obtained by B.R. Choe, H. Koo and H. Yi [CKY, Theorem 2.1] for R1B.

It is worth mentioning that H. Kang and Koo proved similar estimates for the harmonic

Bergman kernel on any smooth bounded domain in Rn [KK, Theorem 1.1]. They adapted

to this setting a method, developed by A. Nagel, J.P. Rosay, E.M. Stein and S. Wainger

[NRSW] in the setting of several complex variables, based on some careful estimates for the

Green operator associated to the Dirichlet problem for the biharmonic equation. It is an

interesting and open question whether this method can be pushed to give sharp estimates

for generalized Bergman kernels on smooth domains of Rn.

Our estimates, proved in Theorem 1.4.3 below, will be the key to prove mapping prop-

erties of the generalised Bergman projections (see Theorem 1.4.7 below) and for later devel-

opements concerning Hardy spaces (see Section 1.5).

For the sake of brevity, it is convenient to set, for every x and y in B1,

ρ(x, y) :=

√1− 2x · y + |x|2 |y|2, θ(x, y) := d(x, ∂B1) + d(y, ∂B1) + |x− y|

and

ξ(x, y) = 1− |x|2 |y|2 .

Define the extended Poisson kernel, by

P (x, y) =∞∑j=0

Zj(x, y) ∀x, y ∈ B1. (1.4.1)

It is known [ABR, Formula 8.11] that

P (x, y) =ξ(x, y)

ρ(x, y)n∀x, y ∈ B1. (1.4.2)


Furthermore [ABR, Theorem 8.13]

RB1(x, y) =ρ(x, y)−n

n c(n)

(n ξ(x, y)2

ρ(x, y)2− 4 |x|2 |y|2

).

We shall need the following technical lemma.

Lemma 1.4.1. The following hold:

(i) for each pair of multiindices α and β there exists a constant C such that∣∣DαxD

βyR

1B1

(x, y)∣∣ ≤ C ρ(x, y)−(n+|α+β|) ∀x, y ∈ B1; (1.4.3)

(ii) for every x and y in B1

1√6θ(x, y) ≤ ρ(x, y) ≤

√2 θ(x, y);

(iii) the polynomial αhj may be written as

αhj (x) =h∑i=0

(−1)iAhi (1− |x|2)i, (1.4.4)

where

Ahi = Ahi (j, n) =

(h

i

)2h−i h . . . (i+ 1)∏2h−1l=h+i(n+ 2j + 2l)

∀i ∈ 0, . . . , h− 1. (1.4.5)

Proof. First we prove (i) in the case where α = β = 0. Observe that

1− 2x · y + |x|2 |y|2 ≥ 1− 2 |x| |y|+ |x|2 |y|2 = (1− |x| |y|)2.

Henceξ(x, y)

ρ(x, y)≤ 1 + |x| |y| ≤ 2 (1.4.6)

and the required estimate for R1B1

follows. We refer to [CKY, Theorem 2.1] for the case

|α + β| > 0.

Next we prove (ii). On the one hand

ρ(x, y)2 = (1− |x|2)(1− |y|2) + |x− y|2

≤ (1 + |x|) (1 + |y|) 1

2

[(1− |x|)2 + (1− |y|)2

]+ |x− y|2

≤ 2 [(1− |x|)2 + (1− |y|)2 + |x− y|2]

≤ 2 θ(x, y)2,


from which the required right hand inequality follows by taking square roots of both sides.

On the other hand

|x− y|2 ≥∣∣|x| − |y|∣∣2 =

∣∣(1− |y|)− (1− |x|)∣∣2.

Therefore

ρ(x, y)2 = (1− |x|2)(1− |y|2) + |x− y|2

= (1− |x|2)(1− |y|2) +1

2|x− y|2 +

1

2|x− y|2

≥ (1− |x|)(1− |y|) +1

2

[(1− |x|)2 + (1− |y|)2 − 2 (1− |x|)(1− |y|)

]+

1

2|x− y|2

=1

2

[(1− |x|)2 + (1− |y|)2 + |x− y|2

]≥ 1

6θ(x, y)2.

This concludes the proof of (ii).

To prove (iii), observe that

αhj (x) =h−1∑m=0

Chm (|x|2 − 1 + 1)m + (|x|2 − 1 + 1)h

=h−1∑m=0

Chm

m∑i=0

(−1)i(m

i

)(1− |x|2)i +

h∑i=0

(−1)i(h

i

)(1− |x|2)i

=h−1∑i=0

(−1)i[ h−1∑m=i

Chm

(m

i

)+

(h

i

)](1− |x|2)i + (−1)h(1− |x|2)h.

(1.4.7)

For the rest of this proof, we set ηl := n+ 2j + 2l. The computations are similar to those in

the proof of Lemma 1.2.6. We claim that, for any N ∈ 1, . . . , h− i− 1,

Ahi(hi

)2N h . . . (h−N + 1)

=h−N−1∑m=i

(−1)h−N−m(h−N − im− i

) ∏h−1l=m+N ηl∏2h−1l=h+m ηl

+1∏2h−1

l=2h−N ηl.

(1.4.8)

We first prove (1.4.8) for N = 1. We recall that Chm = (−1)h−m

(hm

)∏h−1l=m ηl/

∏2h−1l=h+m ηl, and(

h

m

)(m

i

)=

(h

i

)1

(m− i)! (h−m)!=

(h

i

)(h− im− i

).

Then,

Ahi =

(h

i

)h−1∑m=i

(−1)h−m(h− im− i

) ∏h−1l=m ηl∏2h−1

l=h+m ηl+ 1

. (1.4.9)


Since(h−i

0

)=(h−i−1

0

), and

(h−im−i

)=(h−i−1m−i

)+(h−i−1m−i−1

)for m ∈ i+ 1, . . . , h− 1,

Ahi =

(h

i

)(−1)h−i

(h− i− 1

0

) ∏h−1l=i ηl∏2h−1l=h+i ηl

+h−1∑

m=i+1

(−1)h−m[(h− i− 1

m− 1

)+

(h− i− 1

m− i− 1

)] ∏h−1l=m ηl∏2h−1

l=h+m ηl+ 1

.

Thus, each term in the sum in (1.4.9), except the one corresponding to m = i, is decom-

posed into the sum of two terms, the first one containing(h−i−1m−1

), the second one containing(

h−i−1m−i−1

). It is easy to see that, for each m ∈ i + 1, . . . , h − 2, the first term from the

decomposition of the term corresponding to m and the second one from the decomposition

of the term corresponding to m + 1 contain the same binomial coefficient. Moreover, they

share∏h−1

l=m+1 ηl/∏2h−1

l=h+m+1 ηl. Similar considerations holds for the term corresponding to

m = i and the second term corresponding to m = i+ 1. We may then sum up by collecting

like terms, and obtain

Ahi =

(h

i

)h−2∑m=i

(−1)h−m−1

(h− i− 1

m− i

) ∏h−1l=m+1 ηl∏2h−1

l=h+m+1 ηl

[1− n+ 2j + 2m

n+ 2j + 2(h+m)

]+1− n+ 2j + 2(h− 1)

n+ 2j + 2(2h− 1)

=

(h

i

)2h

h−1∑m=i

(−1)h−m−1

(h−m− 1

m− i

) ∏h−1l=m+1 ηl∏2h−1l=h+m ηl

+1

n+ 2j + 2(2h− 1)

,

which is (1.4.8) with N = 1. Next, we assume that (1.4.8) holds for a certain N and

we prove it for N + 1. Similarly as before, we write(h−i−N−1

0

)instead of

(h−i−N

0

), and

we decompose(h−i−Nm−i

)as[(h−i−N−1

m−i

)+(h−i−N−1m−i−1

)]. Then, for each m ∈ i, . . . , k − N −

2, there are two terms containing(h−i−N−1

m−i

), and they also share the common factor∏h−1

l=m+N+1 ηl/∏2h−1

l=h+m+1 ηl. We sum up, yielding

Ahi(hi

)2N h . . . (h−N + 1)

=h−N−2∑m=i

(−1)h−N−m−1

(h−N − i− 1

m− i

) ∏h−1l=m+N+1 ηl∏2h−1l=h+m+1 ηl

[1− n+ 2j + 2(m+N)

n+ 2j + 2(h+m)

]+

1∏2h−1l=2h−N ηl

[1− n+ 2j + 2(h− 1)

n+ 2j + 2(2h−N − 1)

]

= 2(h−N) h−N−2∑

m=i

(−1)h−N−m−1

(h−N − i− 1

m− i

) ∏h−1l=m+N+1 ηl∏2h−1l=h+m ηl

+1∏2h−1

l=2h−N−1 ηl

.


This concludes the proof of (1.4.8). In particular, (1.4.8) with N = h− i− 1 gives

Ahi =

(hi

)2h−i−1 h . . . (i+ 2)∏2h−1

l=h+i+1 ηl

[1− n+ 2j + 2(h− 1)

n+ 2j + 2(h+ i)

]=

(hi

)2h−i h . . . (i+ 1)∏2h−1

l=h+i ηl,

as required.

Lemma 1.4.2. Suppose that (α, β) is a pair of multi-indices. The following hold:

(i) there exists a positive constant C such that∣∣DαxD

βyP (x, y)

∣∣ ≤ C ρ(x, y)−(n−1+|α+β|) ∀x, y ∈ B1;

(ii) for every integer n ≥ 2 and for every nonnegative integer ν there exists a constant C

such that∣∣∣DαxD

βy

∞∑j=0

πν(j) Zj(x, y)∣∣∣ ≤ C ρ(x, y)−(n+ν−1+|α+β|) ∀x, y ∈ B1 (1.4.10)

where πν(j) = 2j (2j − 1) . . . (2j − ν + 1).

Proof. First we prove (i). Recall formula (1.4.2) for the Poisson kernel. Since P is smooth

in B1 ×B1, the required estimate is trivial when ρ(x, y) ≥ 1/2, so that we may assume that

ρ(x, y) < 1/2. In particular, x and y are both away from the origin. By (1.4.6),

ξ(x, y) ≤ 2 ρ(x, y).

Since ξ is a polynomial in x and y, the trivial estimate∣∣DαxD

βy ξ(x, y)

∣∣ ≤ C ∀x, y ∈ B1

holds for every pair of multi-indices α and β such that |α + β| ≥ 1. Note also that there

exists a constant C such that∣∣DαxD

βyρ−n(x, y)

∣∣ ≤ C ρ(x, y)−(n+|α+β|).

(see [CKY, Lemma 2.1]). Therefore, by Leibnitz’s rule,∣∣DαxD

βyP (x, y)

∣∣ ≤ ∑α′+α′′=α

∑β′+β′′=β

∣∣Dα′

x Dβ′

y ξ(x, y)∣∣ ∣∣Dα′′

x Dβ′′

y ρ−n(x, y)

∣∣≤ C

∑α′+α′′=α

∑β′+β′′=β

ρ(x, y)1−|α′+β′| ρ(x, y)−n−|α′′+β′′|,


and (i) is proved.

To prove (ii) we argue as in [ABR, page 179] and write∣∣∣DαxD

βy

∞∑j=0

πν(j) Znj (x, y)

∣∣∣ =∣∣∣Dα

xDβy

∞∑j=0

∂νt[t2jZn

j (x, y)]|t=1

∣∣∣=∣∣∣Dα

xDβy∂

νt

∞∑j=0

Znj (tx, ty)∣∣t=1

∣∣∣=∣∣Dα

xDβy∂

νt

[P (tx, ty)

]|t=1

∣∣≤ C

∑|γ+δ|=ν

∣∣Dα+γx Dβ+δ

y P (x, y)∣∣

≤ C ρ(x, y)−(n+ν−1+|α+β|),

where the last inequality follows from (i). This concludes the proof of (ii) and of the lemma.

We are now in position to prove the main theorem of this section.

Theorem 1.4.3. Let B be an open ball in Rn. For any pair (α, β) of multi-indices there

exists a constant C such that∣∣DαxD

βyR

kB(x, y)

∣∣ ≤ C ρ(x, y)−(n+|α+β|) ∀x, y ∈ B. (1.4.11)

Proof. By Proposition 1.3.4 and Lemma 1.4.1, it remains to estimate the derivatives of RMhB1

,

h = 1, . . . , k − 1. By (1.3.7) and (1.4.4)

RMhB1

(x, y) =1

n c(n)

∞∑j=0

σhj

h∑i1=0

Ahi1(j, n) (1− |x|2)i1Zj(x, y)h∑

i2=0

Ahi2(j, n) (1− |y|2)i2 .

Thus, RMhB1

(x, y) is a finite linear combination of terms of the form

(1− |x|2)i1 (1− |y|)i2∞∑j=0

σhj Ahi1

(j, n)Ahi2(j, n) Zj(x, y) i1, i2 ∈ 0, . . . , h.

Since RMhB1

is h-harmonic in each variable, it is smooth in B1×B1, hence (1.4.11) is trivially

satisfied when x or y are far from ∂B1. Therefore, we may assume that both x and y are

close to ∂B1. By (1.3.6) and (1.4.5), σhj Ahi1

(j, n)Ahi2(j, n) is a polynomial of degree i1 + i2 +1.

Since the polynomials

1, π1(s), . . . , πν(s)


form a basis for the vector space of all polynomials of degree at most ν, there exist constants

d0, . . . , di1+i1+1 such that

σhj Ahi1

(j, n)Ahi2(j, n) =

i1+i1+1∑`=0

d` π`(j).

Thus the problem of estimating RMhB1

is reduced to that of estimating terms of the form

Ti1,i2,`(x, y) := (1− |x|2)i1 (1− |y|)i2∞∑j=0

π`(j) Zj(x, y), (1.4.12)

where i1 and i2 are in 0, . . . , h and ` ≤ i1 + i2 + 1. Note that∣∣Dαx (1− |x|2)i1

∣∣ ≤ C min(ρ(x, y)i1−|α|, 1

)∀x, y ∈ B1, (1.4.13)

and that a similar estimate holds for the derivatives of (1− |y|2)i2 . Indeed, if |α| ≥ i1, then∣∣Dαx (1− |x|2)i1

∣∣ is uniformly bounded in B1, and if |α| < i1, then∣∣Dαx (1− |x|2)i1

∣∣ ≤ C (1− |x|)i1−|α| ≤ C θ(x, y)i1−|α|

and the required estimate follows from Lemma 1.4.1 (ii).

Finally, by Leibnitz’s rule and estimates (1.4.10) and (1.4.13),∣∣∣DαxD

βyTi1,i2,`(x, y)

∣∣∣≤

∑α′+α′′=α

∑β′+β′′=β

∣∣Dα′

x (1− |x|2)i1 Dβ′

y (1− |y|)i2−1∣∣ ∣∣∣Dα′′

x Dβ′′

y

∞∑j=0

π`(j)Zj(x, y)∣∣∣

≤ C ρ(x, y)i1−|α′| ρ(x, y)i2−|β

′| ρ(x, y)−(n+`−1+|α′′+β′′|)

≤ C ρ(x, y)−(n+|α+β|),

as required to complete the proof of the estimates for RkB1

.

The required estimates for a generic ball B follow from the estimates above and formula

(1.3.4).

In the last part of this section we prove some interesting consequences of Theorem 1.4.3.

Definition 1.4.4. Let B be an open ball in Rn. The orthogonal projection PkB of L2(B)

onto b2k(B) is called the k-harmonic Bergman projection on B.


Remark 1.4.5. The k-harmonic Bergman projection is given by

PkBu(x) =

∫B

RkB(x, y)u(y) dy ∀x ∈ B. (1.4.14)

Indeed, on the one hand, we already know that PkB, restricted to b2

k(B) is the identity

operator, for RkB is the reproducing kernel of b2

k(B).

On the other hand, suppose that v is in L2(B) and it is orthogonal to b2k(B). Then, in par-

ticular, v is orthogonal to all k-harmonic polynomials, hence to RkB(x, ·), for the reproducing

kernel is a combination of k-harmonic polynomials (see Propositions 1.3.4 (ii) and 1.3.5).

Remark 1.4.6. Notice that the space of k-harmonic polynomials is dense in bpk(B) for all p

in [1,∞). First observe that it suffices to prove the result in the case where B = B1. Next,

let u be in bpk(B1). Then its r-dilate ur, defined by

ur(x) = u(rx) ∀x ∈ (1/r)B1,

is in bpk((1/r)B1). Moreover, ur is k-harmonic and bounded in a neighbourhood of B1. Since

ur tends to u in bpk(B1), it suffices to prove that k-harmonic functions which are bounded in

a neighbourhood of B1 may be approximated in the Lp norm by k-harmonic polynomials.

Let v be any such function. By [ABR, Corollary 5.34], v may be approximated uniformly in

a neighbourhood of B1, hence in the Lp norm.

It is natural to speculate whether PkB extends to a bounded operator on Lp(B) for some

p ∈ (1,∞). It is known that P1B possesses this property. It may be interesting to notice

that the analogous property for general domains Ω (in place of B) in Rn is false in general.

Indeed, there are starlike domains Ω in Rn with sharp intruding corners for which P1Ω fails to

extend to a bounded operator on Lp(Ω) for some p 6= 2 [CC]. Furthermore, for these starlike

domains the Banach dual of bp1(Ω) fails to be bp′

1 (Ω). Here p′ denotes the index conjugate

to p. Set

bp′

k (B)⊥ :=v ∈ bpk(B) :

∫B

g(x) v(x) dx = 0 for all g ∈ bp′

k (B)

(1.4.15)

The following result holds.

Theorem 1.4.7. Let B denote an open ball in Rn and suppose that p is in (1,∞). Then

(i) the k-harmonic Bergman projection PkB extends to a bounded operator on Lp(B);


(ii) the Banach dual of bpk(B) is bp′

k (B), where p′ denotes the index conjugate to p;

(iii) for every f in Lp(B) there exist unique functions u in bpk(B) and v in bp′

k (B)⊥ such

that f = u+ v. Furthermore,

u = PkBf and v = (I −Pk

B)f.

Proof. Part (i) may be obtained by arguing much as in the proof of [KK, Theorem 4.2]. We

omit the details.

Next we prove (ii). Clearly, any function in bp′

k (B) represents a continuous linear func-

tional on bpk(B).

Conversely, consider a continuous linear functional λ on bpk(B). By the Hahn–Banach

theorem, λ has an extension λ to a continuous linear functional on Lp(B). Denote by fλ the

function in Lp′(B) that represents λ. From (i) we deduce that Pk

Bfλ is in bp′

k (B). Clearly,

for every k-harmonic polynomial v

λ(v) =

∫B

v(x) fλ(x) dx

=

∫B

fλ(x) dx

∫B

RkB(x, y) v(y) dy.

By Theorem 1.4.3, for every y in B the function RkB(·, y) is in Lp(B). Therefore, by Fubini’s

theorem, we may interchange the order of integration and obtain

λ(v) =

∫B

v(y) dy

∫B

fλ(x)RkB(x, y) dx

=

∫B

v(y) PkBfλ(y) dy.

Thus, the restriction of λ to the space of k-harmonic polynomials is represented by PkBfλ.

Since the space of k-harmonic polynomials is dense in bpk(B) by Remark 1.4.6, the function

PkBfλ represents λ on bpk(B). This proves (ii).

Part (iii) is a routine consequence of (ii). We omit the details.

Corollary 1.4.8. Let p be in (1,∞), and denote by p′ its conjugate index. There exists a

constant C such that for every ball B in Rn

‖RkB(x, ·)‖p ≤ C d(x, ∂B)−n/p

′ ∀x ∈ B (1.4.16)

and

‖DαxR

kB(x, ·)‖p ≤ C d(x, ∂B)−|α|−n/p

′ ∀x ∈ B. (1.4.17)


Proof. By the Hahn–Banach Theorem, the evaluation functional Λx on bpk(B) extends to a

continuous linear functional Λx on Lp(B) such that |||Λx|||Lp(B) = |||Λx|||bpk(B). Denote by h

the function in Lp′(B) that represents Λx. By Theorem 1.4.7 (iii) (with the role of p and p′

interchanged) h = PkBh + (I −Pk

B)h, with PkBh in bp

′

k (B) and (I −PkB)h in bpk(B)⊥. It

is straightforward to check that PkBh represents Λx, whence Pk

Bh = RkB(x, ·). Since Pk

B is

bounded on Lp′(B) by Theorem 1.4.7 (i),

‖RkB(x, ·)‖p′ ≤

∣∣∣∣∣∣PkB

∣∣∣∣∣∣p′‖h‖p′

=∣∣∣∣∣∣Pk

B

∣∣∣∣∣∣p′

∣∣∣∣∣∣Λx

∣∣∣∣∣∣Lp(B)

=∣∣∣∣∣∣Pk

B

∣∣∣∣∣∣p′

∣∣∣∣∣∣Λx

∣∣∣∣∣∣bpk(B)

≤ C d(x, ∂B)−n/p′,

as required.

The estimate (1.4.17) is proved similarly. Indeed, a straightforward consequence of the

reproducing formula (1.3.2) is that

Dαu(x) =

∫B

DαxR

kB(x, y)u(y) dy ∀u ∈ b2

k(B).

Thus, the continuous linear functional u ∈ bp′

k (B) 7→ Dαu(x) is represented by DαxR

kB(x, ·),

and the required estimate follows from (1.3.1).

1.5 Application to Hardy spaces

We recall the definition of the atomic Hardy space H1(Rn).

Definition 1.5.1. Suppose that 1 < p ≤ ∞. An H1,p-atom a is a function in Lp(Rn),

supported in a ball B ∈ B, with the following properties:

(i)∫Ba(x) dx = 0;

(ii) ‖a‖p ≤ |B|−1/p′ .

Definition 1.5.2. The Hardy space H1,p(Rn) is the space of all functions f in L1(Rn) that

admit a decomposition of the form

f =∞∑j=1

cj aj, (1.5.1)

1.5. APPLICATION TO HARDY SPACES 33

where aj are H1,p-atoms, and∑∞

j=1 |cj| < ∞. The norm ‖f‖H1 of f is the infimum of∑∞j=1 |cj| over all decompositions (2.3.1) of f .

It is a beautiful result of Coifman and Weiss [CW, Theorem A, p. 592] that all the spaces

H1,p(Rn) agree, when 1 < p ≤ ∞, and the corresponding norms are equivalent. We will

simply denote them all by H1(Rn), endowed with any of the equivalent norms above.

We now define special atoms, which satisfy, instead of (i) above, the much stronger

cancellation condition of being orthogonal to the generalised harmonic Bergman space bpk(B).

Definition 1.5.3. Suppose that k is a positive integer, p is in (1,∞) and denote by p′ its

conjugate index. A special k-atom in Lp (or Xk,p-atom) associated to the open ball B is a

function A in Lp(B) such that

(i)∫BA(x) q(x) dx = 0 for each k-harmonic polynomial q;

(ii) ‖A‖p ≤ |B|−1/p′ .

Remark 1.5.4. Note that condition (i) implies that∫BA(x) dx = 0. Thus, an Xk,p-atom is

an H1-atom.

Definition 1.5.5. We define Xk,p(Rn) as the space of all functions in L1(Rn) which admit

a decomposition of the form

f =∞∑j=1

cj Aj, (1.5.2)

where Aj are Xk,p-atoms, and∑∞

j=1 |cj| <∞. We define

‖f‖Xk,p = inf∑j

|λj|, (1.5.3)

the infimum being taken over all representations of f of the form (1.5.2).

Lemma 1.5.6. Let p ∈ (1,∞). There exists a constant C such that for every H1-atom a

there exist a summable sequence of complex numbers cj and a sequence Aj of Xk,p-atoms

such that

a =∑j

cj Aj and∑j

|cj| ≤ C.


Proof. The translation x 7→ x0 + x maps an atom associated to a ball with centre x0 and

radius t to an atom associated to a ball with centre 0 and radius t. Hence we may assume

that B is a ball with centre 0.

First, we assume that that the support of the atom a is contained in the unit ball B1.

We denote by Bj the ball with centre 0 and radius 2j−1 and by Pkj the projection onto

b2k(Bj). Sometimes it will be convenient to write Pk

0 for I , the identity operator. We recall

that, by Theorem 1.4.7, each Pkj extends to a bounded operator on Lp(B1). Define

Aj =Pk

j−1a−Pkj a

|Bj|1/p′ ‖Pkj−1a−Pk

j a‖p.

Clearly the support of Aj is contained in Bj and

‖Aj‖p ≤ |Bj|−1/p′ .

Observe also that Pkj a = Pk

j (Pkj−1a). Now suppose that q is a k-harmonic polynomial

in Rn. Then∫Bj

[Pk

j−1a(x)−Pkj a(x)

]q(x) dx =

∫Bj

(I −Pkj )(Pk

j−1a)(x) q(x) dx

By Theorem 1.4.7 (iii), (I −Pkj )(Pk

j−1a) is in (bp′

k (Bj))⊥, so that the last integral vanishes.

Thus, Aj is an Xk,p-atom with support contained in Bj. At least formally, we may write

a = a−Pk1 a+

∞∑j=2

[Pk

j−1a−Pkj a]

=∞∑j=1

cj Aj,

where cj = |Bj|1/p′ ‖Pj−1a−Pja‖p. To conclude the proof of the lemma, it suffices to

show that∑∞

j=1 |cj| ≤ C, where C does not depend on the atom a. We denote by Rkj the

reproducing kernel of b2k(Bj). Note that

Pkj a(x) =

∫B1

a(y)Rkj (x, y) dy

=

∫B1

a(y)[Rkj (x, y)−Rk

j (x, 0)]

dy

=

∫ 1

0

dt

∫B1

a(y)∇yRkj (x, ty) · y dy.

1.5. APPLICATION TO HARDY SPACES 35

Hence the generalised Minkowski inequality and (1.4.17) imply that

‖Pkj a‖p ≤

∫ 1

0

dt

∫B1

|a(y)| ‖∇yRj(·, ty)‖p |y| dy

≤ C

∫B1

|a(y)| d(0, ∂Bj)−1−n/p′ dy

≤ C 2−(1+n/p′)j.

(1.5.4)

Therefore

|cj| ≤ C 2−j,

with C independent of a, so that the sequence cj is summable, as required to conclude the

proof of the lemma.

Next, suppose that t is positive and that a is an H1-atom with support contained in

B(0, t). It is straightforward to check that the function a1/t, defined by

a1/t(x) = tna(tx),

is an H1-atom with support contained in B1. Now,

Pkj (a1/t)(x) =

∫B1

tn a(ty)Rkj (x, y) dy

=

∫B(0,t)

a(y)Rkj (x, y/t) dy

= tn∫B(0,t)

a(y) t−nRkj (tx/t, y/t) dy

= tnPktBj

(a)(tx)

=[Pk

tBj(a)]

1/t(x),

so that

PktBj

(a) =[Pk

Bj(a1/t)

]t.

Then ∥∥PktBj

(a)∥∥p

= t−n/p∥∥Pk

Bj(a1/t)

∥∥p

≤ C t−n/p 2−(1+n/p′)j,

where we have used the estimate (2.6.4) in the proof of the lemma. Therefore∣∣tBj

∣∣1/p′ ∥∥PktBj−1

(a)−PktBj

(a)∥∥p≤ C (t2j)n/p

′t−n/p 2−(1+n/p′)j

≤ C 2−j,

and we proceed as in the case of B1.


By Lemma 1.5.6, ‖f‖Xk,p is finite for every f in H1(Rn). Moreover, observe that

‖f‖H1 ≤ ‖f‖Xk,p , (1.5.5)

for every Xk,p-atom is also an H1-atom. It is straightforward to check that f 7→ ‖f‖Xk,p is

a norm on H1(Rn).

Theorem 1.5.7. Every function f in H1(Rn) admits a decomposition of the form

f =∑j

λjAj,

where λj is a summable sequence and the Aj are Xk,p-atoms. Furthermore, there exists a

positive constant c such that

c ‖f‖Xk,p ≤ ‖f‖H1 ≤ ‖f‖Xk,p ∀f ∈ H1(Rn).

Proof. The right hand inequality has already been proved in (1.5.5). Then the identity map

ι is continuous from H1(Rn), endowed with the topology induced by the norm ‖·‖Xk,p , to

H1(Rn), endowed with the topology induced by the norm ‖·‖H1 . Clearly ι is bijective, so

that ι−1 is continuous, i.e., the left hand inequality holds.

Chapter 2

Bergman and Hardy spaces on

Riemannian manifolds

2.1 Basic definitions and background material

We consider a connected noncompact Riemannian manifold M with Riemannian measure µ

and Laplace–Beltrami operator L . We denote by B the family of all balls in M . For each

B in B we denote by cB and rB the centre and the radius of B respectively. Furthermore,

we denote by cB the ball with centre cB and radius c rB.

We shall assume throughout the following:

(i) M possesses the volume doubling property, i.e., there exists a positive constant D0 such

that

µ(2B) ≤ D0 µ(B) ∀B ∈ B; (2.1.1)

(ii) there exist positive constants b and ν such that the relative Faber–Krahn inequality

λ1(U) ≥ b

r2B

(µ(B)

µ(U)

)2/ν

(2.1.2)

holds for any B ∈ B and for any relatively compact open set U ⊂ B. Here λ1(U)

denotes the bottom of the spectrum of the Dirichlet Laplacian LU on U .

In Rn and in many applications, ν is the dimension of M . It is known [Gr1] that every com-

plete noncompact manifold with nonnegative Ricci curvature admits a relative Faber–Krahn

37

38 CHAPTER 2. BERGMAN AND HARDY SPACES ON RIEMANNIAN MANIFOLDS

inequality. An important result [Gr2, p. 410] states that, under assumption (i) above, (ii) is

equivalent to the following diagonal upper estimate for the heat kernel htt>0 associated to

the Laplace-Beltrami operator on M

(DUE) ht(x, x) ≤ C

µ(B(x,

√t)) ∀x ∈M ∀t > 0.

We now describe some noteworthy consequences of the relative Faber–Krahn inequality

(2.1.2) concerning mean value inequalities for L -harmonic functions. Recall that a L -

harmonic function (or simply a harmonic function for short) in an open set Ω is a smooth

function u such that L u = 0 in Ω. These inequalities will be used in the next section in

the study of Bergman spaces and will be the key to obtain estimates for the corresponding

Bergman projections.

Preliminarily, we recall that a subsolution of the heat equation in I × Ω, where I is an

interval in the real line and Ω is an open subset of M , is a real function u in C 2(I ×Ω) such

that∂u

∂t≤ L u.

Theorem 2.1.1. Suppose that B is a relatively compact ball in M which admits a Faber–

Krahn inequality

λ1(U) ≥ a µ(U)−2/ν (2.1.3)

for some positive constants a, ν and for any open subset U of B. Then there exists a constant

C, which depends only on ν, such that the following hold:

(i) for any T > 0 and for any subsolution u(t, x) of the heat equation in the cylinder

C = (0, T ]×B

u+(T, cB)2 ≤ C a−ν/2

min(√T , rB)ν+2

∫C

u2+(t, x) dt dµ(x); (2.1.4)

(ii) for every L -harmonic function u on B

|u(cB)|2 ≤ C a−ν/2

rνB

∫B

|u|2 dµ. (2.1.5)

Proof. A proof of (i) may be found in [Gr1, Gr2].

To prove (ii) observe that since u is L -harmonic, u+ is L -subharmonic. Similarly, since

−u is L -harmonic, (−u)+, which is equal to u−, is L -subharmonic. Then we may apply (i)

to both u+ and u−, and the required estimate follows by setting T = r2B.

2.1. BASIC DEFINITIONS AND BACKGROUND MATERIAL 39

Theorem 2.1.2. Suppose that M possesses the doubling property (2.1.1) and that the relative

Faber–Krahn inequality (2.1.2) holds. Then there exists a constant C such that the following

hold:

(i) for every ball B and every L -harmonic function u in B

|u(cB)|2 ≤ C

µ(B)

∫B

|u|2 dµ; (2.1.6)

(ii) for every ball B and every L -harmonic function u in B

|u(cB)| ≤ C

µ(B)

∫B

|u| dµ. (2.1.7)

Proof. Observe that (i) follows simply by inserting a = b r−2B µ(B)2/ν in (2.1.5).

Part (ii) is essentially due to Li and Wang [LW]. For the sake of completeness we give

full details of the proof. As in [LW], we use an inductive argument.

Set Q := µ(B)−1∫B|u| dµ. By the L2-mean value inequality applied to u on (1/2)B and

by the doubling property (2.1.1),

|u(cB)|2 ≤ C

µ(2−1B)

∫2−1B

|u|2 dµ

≤ C

µ(2−1B)sup2−1B

|u|∫

2−1B

|u| dµ

≤ CQµ(B)

µ(2−1B)sup2−1B

|u|

≤ CQD0 sup2−1B

|u| .

(2.1.8)

For each positive integer k, set Rk :=∑k

j=1 2−j and Sk := supBk |u|, where Bk denotes the

ball Rk B. Note that Bk is an increasing sequence of balls, which contain B1 and approach

asymptotically B.

We claim that

|u(cB)| ≤ (CQ)Rk D∑ki=1(i+1)2−i

0 S2−k

k , (2.1.9)

where C is the same constant as in (2.1.8).

By (2.1.8), the claim holds for k = 1.

Assume that (2.1.9) holds for k. Choose x in Bk such that

|u(x)| = supBk

|u| .


Observe that

B(x, 2−k−1rB) ⊂ Bk+1 ⊂ B ⊂ B(x, 2rB).

By the L2-mean value property (2.1.6) applied to u on B(x, 2−k−1rB) and the doubling

property,

S2k = |u(x)|2 ≤ C

µ(B(x, 2−k−1rB))

∫B(x,2−k−1rB)

|u|2 dµ

≤ C

µ(B(x, 2−k−1rB))sup

B(x,2−k−1rB)

|u|∫B(x,2−k−1rB)

|u| dµ

≤ C

µ(B(x, 2−k−1rB))supBk+1

|u|∫B

|u| dµ

≤ CDk+20

µ(B)

µ(B(x, 2rB))QSk+1

≤ C QDk+20 Sk+1.

This, together with (2.1.9), gives

|u(cB)| ≤ (CQ)Rk D∑ki=1(i+1)2−i

0

[C QDk+2

0 Sk+1

]2−k−1

= (CQ)Rk+1 D∑k+1i=1 (i+1)2−i

0 S2−k−1

k+1 ,

which is (2.1.9) for k + 1.

The required L1-mean value inequality follows by taking the limit of both sides of (2.1.9)

as k tends to ∞.

2.2 Harmonic Bergman spaces

In this section we define the Hardy-type spaces we shall study in the rest of this chapter.

Their definition involves the so-called harmonic Bergman spaces.

Definition 2.2.1. For every p ∈ [1,∞) and for every open subset Ω of M , the Bergman

space bp(Ω) is the space of all harmonic functions in Lp(Ω), i.e., the space of all functions H

in Lp(Ω) such that LH = 0 on Ω.

The Bergman space bp(Ω), endowed with the Lp(Ω) norm, is a closed subspace of Lp(Ω),

hence a Banach space. Indeed, given a Cauchy sequence fn in bp(Ω), there exists a function

2.2. HARMONIC BERGMAN SPACES 41

f in Lp(Ω) such that ‖f − fn‖p is convergent to 0. To prove that f is harmonic, observe

that for every smooth function ϕ with compact support contained in Ω

〈ϕ,L f〉 = 〈Lϕ, f〉

= limn→∞

〈Lϕ, fn〉

= limn→∞

〈ϕ,L fn〉

= 0.

Hence L f = 0 in Ω in the sense of distributions. By elliptic regularity f is smooth, whence

L f = 0 pointwise and f is harmonic.

The spaces bp(Ω) have been studied in various settings. For basic properties of bp(Ω)

when Ω ⊂ Rn, and in particular when Ω is an Euclidean ball, see [ABR, CKY] and the first

chapter of this thesis. Other interesting results are contained in [KK], where estimates for

the Bergman kernel on smooth domains in Rn are established.

A noteworthy consequence of the L1-mean value inequality is that the evaluation func-

tional λx at a point x of a domain Ω, i.e., the linear functional λx(u) = u(x), is continuous

on bp(Ω) for any p ∈ [1,∞).

Proposition 2.2.2. There exists a constant C, independent of x in Ω and p in [1,∞), such

that

|||λx|||bp(Ω) ≤C1/p

µ(B(x,R)

)1/p∀u ∈ bp(Ω),

where R denotes the distance of x from ∂Ω.

Proof. Indeed, denote by R the distance of x from ∂Ω. By Theorem 2.1.2 (ii) and Holder’s

inequality,

|u(x)| ≤[ C

µ(B(x,R)

) ∫B(x,R)

|u|p dµ]1/p

≤ C1/p

µ(B(x,R)

)1/p‖u‖bp(Ω) ∀u ∈ bp(Ω),

(2.2.1)

where C is the constant appearing in (2.1.7), which is independent of u in bp(Ω), x in Ω and

p in [1,∞). The required estimate of |||λx|||bp(Ω) follows.

In the case where p = 2, by the Riesz representation theorem, there exists a unique

function RΩ(x, ·) in b2(Ω) that represents the functional λx, i.e.,

u(x) =

∫Ω

RΩ(x, y)u(y) dµ(y) ∀u ∈ b2(Ω).


The function RΩ is called the Bergman kernel for the domain Ω. Quite a few properties

of RΩ may be established by abstract nonsense. We refer the reader to the classical paper

of N. Aronszajn [A] for a nice exposition of the theory of reproducing kernels. Recall that

b2(Ω) is a closed subspace of L2(Ω). The orthogonal projection of L2(Ω) onto b2(Ω) is called

the Bergman projection and will be denoted by PΩ. Various properties of PΩ will play an

important role in the sequel. We collect them in the following proposition.

Proposition 2.2.3. Suppose that M possesses the doubling property (2.1.1) and that the

relative Faber–Krahn inequality (2.1.2) holds. Let Ω be a bounded domain in M . Then the

following hold:

(i) the function RΩ is real-valued and RΩ(x, y) = RΩ(y, x) for every x, y in Ω. Further-

more, RΩ(x, ·) is in b2(Ω);

(ii) the orthogonal projection PΩ of L2(Ω) onto b2(Ω) is given by

PΩf(x) =

∫Ω

RΩ(x, y) f(y) dµ(y) ∀f ∈ L2(Ω);

(iii) ‖RΩ(x, ·)‖b2(Ω) = RΩ(x, x)1/2 for every x in Ω.

Proof. The proof of (i) and (ii) is classical and may be found in [A]. The proof of (iii) is

straightforward and may be found in [ABR, Chapter 8].

Observe that we may write

PΩf(x) =

∫Ω

RΩ(x, y) f(y) dµ(y) ∀f ∈ Lp(Ω) ∩ L2(Ω).

It is natural to speculate whether the projection PΩ extends to a bounded operator on bp(Ω)

for p in (1,∞) and the Banach dual of bp(Ω) is bp′(Ω). Here p′ denotes the index conjugate

to p. In Section 1.4, we have already observed that this is not always the case even in Rn

[CC]. We now prove that these pathologies disappear if we consider domains Ω with smooth

boundary.

Theorem 2.2.4. Suppose that M possesses the doubling property (2.1.1) and that the rel-

ative Faber–Krahn inequality (2.1.2) holds. Let Ω be a bounded domain in M with smooth

boundary. Then the following hold:

(i) the Bergman projection extends to a bounded operator on Lp(Ω) for all p in (1,∞);


(ii) the Banach dual of bp(Ω) is bp′(Ω), where p′ denotes the index conjugate to p.

(iii) for every f in Lp(Ω) there exist unique functions u in bpk(Ω) and v in bp′

k (Ω)⊥ such that

f = u+ v. Furthermore,

u = PkΩf and v = (I −Pk

Ω)f.

Proof. The proof of (i) may be deduced from the biharmonic equation approach using stan-

dard results in elliptic theory (see also [CC, p. 700] and the references therein). The result

is contained in [Me1]. The proofs of (ii) and (iii) follow the same lines as the proofs of the

corresponfing results in the Euclidean case (see Theorem 1.4.7), and are omitted.

If R < Injp(M), then ∂B(p,R) is a smooth hypersurface in M . Unfortunately, larger open

balls in M may not have smooth boundary (think of the case of a cylinder in R3), so that the

theorem above may not be applicable to (some) large balls. For later developments, we shall

need to work with open domains, which resemble open balls, but have smooth boundary.

Here is the precise definition.

Definition 2.2.5. Suppose that R and ε are positive numbers and p is a point in M . A

connected open subset Ω of M with smooth boundary is said to be an approximate (R, ε)-

ball with centre p if there exist two balls B and B′ with centre p and radii R and (1 + ε)R,

respectively, such that

B ⊆ Ω ⊆ B′.

Clearly, any open ball with smooth boundary and radius R is an approximate (R, ε)-ball for

every ε > 0. A basic question is whether approximate (R, ε)-balls with centre p exist for

every positive R and ε and for every p in M . The following proposition answers the question

in the positive.

Proposition 2.2.6. Suppose that R and ε are positive numbers and p is a point in M . Then

there exists infinitely many approximate (R, ε)-balls with centre p.

Proof. It suffices to prove the result for ε small. Denote by B and 2B the balls with centre

p and radius R and 2R, respectively. Since M is assumed to be complete, 2B is compact.

Denote by ρε/100

the Gaffney regularised distance with base point p such that∣∣∣ρε/100(x)− d(x, p)∣∣∣ ≤ 10−2 ε ∀x ∈ 2B.


Such a distance exists and it is smooth, as shown by M. Gaffney in [Ga]. By Sard’s theorem

(see, for instance, [Au, Theorem 1.30]), the set of critical values of ρε/100

is of null measure

in((1 + ε/3)R, (1 + 2ε/3)R

). Suppose that c is a noncritical value in this interval, and set

Ω := x ∈ 2B : ρε/100

(x) < c.

Then ∂Ω is a smooth hypersurface in M . It is straighforward to check that B ⊂ Ω ⊂ (1+ε)B,

so that Ω is an approximate (R, ε)-ball with centre p, as required.

Proposition 2.2.7. Suppose that M possesses the doubling property (2.1.1) and that the

relative Faber–Krahn inequality (2.1.2) holds. Suppose that c1 and c2 are numbers such that

1 < c1 < c2. Then the following hold:

(i) for each p in [1,∞) there exists a constant C, independent of B in B such that for

every c in (c1, c2) and for every approximate (crB, c2− c)-ball Ω with centre cB and for

every x in B

|u(x)| ≤ C µ(B)−1/p ‖u‖p ∀u ∈ bp(Ω);

(ii) for each p in (1,∞) there exists a constant C, independent of B in B such that for

every c in (c1, c2) and for every approximate (crB, c2 − c)-ball Ω with centre cB[∫Ω

∣∣RΩ(x, y)∣∣p′ dµ(y)

]1/p′

≤ C µ(B)−1/p ∀x ∈ B, (2.2.2)

where p′ denotes the index conjugate to p;

(iii) there exists a constant C, independent of B in B such that for every c in (c1, c2) and

for every approximate (crB, c2 − c)-ball Ω with centre cB

supy∈Ω

∣∣RΩ(x, y)∣∣ ≤ C µ(B)−1 ∀x ∈ B; (2.2.3)

(iv) the projection operator PΩ, is bounded from Lp(B) to L∞(Ω) and

supB∈B

µ(B)1/p∣∣∣∣∣∣PΩ

∣∣∣∣∣∣Lp(B);L∞(Ω)

<∞.

Consequently, PΩ is bounded from Lp(B) to Lp(Ω) and

supB∈B

∣∣∣∣∣∣PΩ

∣∣∣∣∣∣Lp(B);Lp(Ω)

<∞.


Proof. We first prove (i). Since x is in B and Ω is an approximate (crB, c − c2)-ball, the

distance of x from ∂Ω is at least (c− 1)rB. By (2.2.1),

|u(x)| ≤ C1/p

µ(B(x, (c− 1)rB)

)1/p‖u‖bp(Ω)

≤ C1/p

µ(B(x, 4rB)

)1/p

µ(B(x, 4rB)

)1/p

µ(B(x, (c− 1)rB)

)1/p‖u‖bp(Ω)

≤ C1/pDk/p0

µ(B(x, 4rB)

)1/p‖u‖bp(Ω) ∀u ∈ bp(Ω),

(2.2.4)

where C is the constant appearing in (2.1.7) and k is an integer such that 2k(c− 1) ≥ 4. We

have used the doubling property and the inclusion Ω ⊂ B(x, 4rB). This proves (i).

Next we prove (ii). Suppose that x is in Ω. Recall that the evaluation functional λx,

defined just above formula (2.2.1), is continuous on bp(Ω). Clearly,

λx(u) = u(x)

=

∫Ω

RΩ(x, y)u(y) dµ(y) ∀u ∈ b2(Ω) ∩ bp(Ω).

We claim that b2(Ω) ∩ bp(Ω) is dense in bp(Ω). Since bp(Ω) ⊂ b2(Ω) when p > 2, it suffices

to consider the case where 1 < p < 2. Then b2(Ω) ∩ bp(Ω) is just b2(Ω). We argue by

contradiction. Suppose that b2(Ω) is not dense in bp(Ω). Then there exists a nontrivial

continuous linear functional λ on bp(Ω) which vanishes on b2(Ω). By Theorem 2.2.4 (ii),

there exists a function ϕ in bp′(Ω) such that∫

Ω

f ϕ dµ = 0 ∀f ∈ b2(Ω).

Since bp′(Ω) is contained in b2(Ω), the formula above implies that ϕ 7→

∫Ωf ϕ dµ is the null

functional on b2(Ω). Hence ϕ = 0, which contradicts the fact that λ 6= 0. Note that here we

use the fact that ∂Ω is smooth.

Thus,

|||λx|||bp(Ω) = sup |λx(u)|

= sup∣∣∣∫

Ω

RΩ(x, y)u(y) dµ(y)∣∣∣,

where the supremum is taken over all u in b2(Ω) ∩ bp(Ω) with ‖u‖p ≤ 1. By arguing much

as in the proof of Corollary 1.4.8, we may prove that there exists a constant, depending on


p, but not on x in Ω such that

‖RΩ(x.·)‖p′ ≤ C∣∣∣∣∣∣λx∣∣∣∣∣∣bp(Ω)

,

By combining this with (2.2.1), we conclude that

‖RΩ(x, ·)‖p′ ≤C1/p

µ(B(x,R)

)1/p,

where C is the constant appearing in (2.1.7), which is independent of x in B and p in (1,∞),

and R is the distance between x and ∂Ω. Since R is at least (c− 1)rB,

µ(B(x,R)

)≥ C µ(B),

by the doubling property. Here C depends on c, but not on B. This concludes the proof

of (ii).

Statement (iii) is contained in [Me1], and follows from careful estimates for the Green

function associated to the biharmonic equation.

To prove (iv), observe that, by (iii),∣∣PΩf(x)∣∣ ≤ ∫

B

∥∥RΩ(·, y)∥∥L∞(Ω)

|f(y)| dµ(y)

≤ C

µ(B)

∫B

|f(y)| dµ(y)

≤ C µ(B)−1/p∥∥f∥∥

p∀f ∈ Lp(B),

where C is independent of x in Ω. The first of the two required estimate follows by taking

the supremum of both sides with respect to x in Ω.

To prove the second estimate, we observe that∥∥PΩf∥∥Lp(Ω)

≤ µ(Ω)1/p∥∥PΩf

∥∥L∞(Ω)

,

and that there exists a constant C, which depends on c1, c2, but not on the ball B, such

that

µ(Ω) ≤ C µ(B).

Therefore, ∥∥PΩf∥∥Lp(Ω)

≤ C1/p µ(B)1/p ‖f‖Lp(B) ∀f ∈ Lp(B),

where C is the same as above, as required.

2.3. HARDY-TYPE SPACES 47

2.3 Hardy-type spaces

Under the standing assumption that the Riemannian measure µ is doubling, M is a space of

homogeneous type in the sense of Coifman and Weiss. We recall the definition of the atomic

Hardy space H1(M).

Definition 2.3.1. Suppose that 1 < p ≤ ∞. An H1,p-atom a is a function in Lp(M), with

support contained in a ball B, with the following properties

(i)∫Ba dµ = 0;

(ii) ‖a‖p ≤ µ(B)−1/p′ (where p′ denotes the index conjugate to p).

Definition 2.3.2. The Hardy space H1,p(M) is the space of all functions f in L1(M) that

admit a decomposition of the form

f =∞∑j=1

cj aj, (2.3.1)

where aj are H1,p-atoms and∑∞

j=1 |cj| < ∞. The norm ‖f‖H1,p of f is the infimum of∑∞j=1 |cj| over all decompositions (2.3.1) of f .

It is a beautiful result of R.R. Coifman and G. Weiss [CW, Theorem A, p. 592] that the

spaces H1,p(M) agree, when 1 < p ≤ ∞, and the corresponding norms are equivalent.

We now define special atoms, which satisfy, instead of (i) above, the much stronger

cancellation condition of being orthogonal to the harmonic Bergman space bp(B). Then a

Hardy-type space is defined as in the classical case of Coifman and Weiss, but with special

atoms in place of classical atoms.

Definition 2.3.3. Suppose that p is in (1,∞) and denote by p′ its conjugate index. A

special atom in Lp (or X1,p-atom) associated to the ball B is a function A in Lp(M), with

support contained in B and such that

(i)∫MAH dµ = 0 for all H in bp

′(B);

(ii) ‖A‖p ≤ µ(B)−1/p′ .

Note that condition (i) implies that∫MA dµ = 0, because 1B is in bp

′(B). Thus, a special

atom is an H1(M)-atom.


Definition 2.3.4. The Hardy-type space X1(M) associated to special atoms is the space of

all functions f that admit a decomposition of the form

f =∞∑j=1

cj aj, (2.3.2)

where aj are X1,p-atoms and∑∞

j=1 |cj| < ∞. The norm ‖f‖X1,p of f is the infimum of∑∞j=1 |cj| over all decompositions (2.3.2) of f .

It is a nontrivial question to determine whether all the Hardy-type spaces X1,p(M), 1 < p <

∞, agree. Here is our result, whose proof hinges on a variant of an idea of Coifman and

Weiss [CW].

Theorem 2.3.5. Suppose that M possesses the doubling property (2.1.1) and that the relative

Faber–Krahn inequality (2.1.2) holds. For each p in (1,∞), the space X1,p(M) agrees with

X1,2(M), and their norms are equivalent.

Proof. Clearly, if 1 < p1 < p2 < ∞, then X1,p2(M) ⊂ X1,p1(M). Thus, it suffices to prove

that the reverse containment holds. Clearly, it suffices to show that if A is a X1,p1-atom,

then A admits a representation of the form

A =∑j

αjaj, (2.3.3)

where each aj is a X1,p2-atom and∑

j |αj| ≤ D, with D indipendent of A. The proof of this

follows the same lines of the proof of the original result of Coifman and Weiss. However,

there are also differences.

Suppose that A is a X1,p1-atom supported in a ball B and set A0 = µ(B)A. Observe that

‖A0‖p1p1 = µ(B)p1‖A‖p1p1 ≤ µ(B)p1−p1/p′1 = µ(B). (2.3.4)

Let α be a positive number and denote by Oα the setx ∈ M : M |A0|p1 (x) > αp1

.

Here M denotes the uncentred Hardy–Littlewood maximal operator. We refer the reader

to [CW] for all basic properties of the Hardy–Littlewood maximal operator on spaces of

homogeneous type. We shall use all the properties we need of M without further reference

to [CW].


Assume that α is so large that Oα is contained in 2B. Then Oα is a bounded open set,

satisfying

µ(Oα) ≤ C0

(‖A0‖p1/α

)p1 ≤ C0α−p1µ(B)

for a certain C0 > 0. Hence, if α is large enough, µ(Oα) < µ(B) ≤ µ(M) and M \ Oα is

nonempty. We may therefore perform a Whitney-type decomposition of Oα; there exists a

collection of balls Bj : j ∈ N, such that

(i)⋃j Bj = Oα;

(ii) B∗∗j ∩Ocα 6= ∅;

(iii) each point in M belongs to, at most, N balls B∗j , i.e., the collection B∗j has the finite

overlapping property. It is important to note that N does not depend on the set Oα,

but depends only on the geometry of M .

Here B∗j and B∗∗j are balls with the same centre as Bj and radii 2rBj and 6rBj , respectively.

Notice that, by (2.1.1), there exists k0 > 0 such that

µ(B∗∗)

µ(B)≤ k0 for any ball B ⊆M.

Denote by Aj the function defined by

Aj =1Bj∑` 1B`

A0.

Clearly the support of Aj is contained in Bj. Let Ωj denote an approximate (rBj , 1)-ball

with centre cBj . Thus, Ωj has smooth boundary and satisfies Bj ⊆ Ωj ⊆ B∗j . Now, define

bj := Aj −PΩj(Aj),

b :=∑j

bj and g := A0 1Ωcα +∑j

PΩj(Aj).

We first show that these equalities are valid in L1(M). Observe that

‖bj‖L1(Ωj) ≤(1 + |||PΩj |||L1(Bj);L1(Ωj)

)‖Aj‖L1(Bj).


Then, by Proposition 2.2.7(iv), there exists C such that∑j

‖bj‖L1(Ωj) ≤ C∑j

‖A0‖L1(Bj)

≤ CN

∫Oα

|A0| dµ

≤ CN‖A0‖1

≤ CN‖A0‖p1 µ(B)1/p′1

≤ CNµ(B).

Note that the decomposition of A0 into a good part and a bad part is different from the

classical, for we use the regularising operators PΩj instead of the usual averaging operator.

Some properties of b and g are the following.

(1) For any x ∈M , |g(x)| ≤ Ck1/p10 αN , where C is the constant appearing in Proposition

2.2.7(iii). Indeed, if x 6∈ Oα, by Lebesgue differentiability theorem

|g(x)| = |b(x)| ≤(M |A0|p1 (x)

)1/p1 ≤ α,

whereas, if x ∈ Oα, g(x) =∑

j PΩj(Aj)(x). Notice that at most N terms in this sum

are non-zero. By Proposition 2.2.7(iv), we then have that

|g(x)| ≤∑j

∣∣PΩj(Aj)(x)∣∣

≤ C∑j

µ(Bj)−1/p1 ‖Aj‖p1

≤ C∑j

(1

µ(B∗∗j )

∫B∗∗j

|A0|p1 dµ

)1/p1 (µ(B∗∗j )

µ(Bj)

)1/p1

≤ Ck1/p10 αN.

(2) supp(g) ⊆ B∗. Indeed, we already observed that Oα ⊆ B∗. On the other hand, g = A0

on Ocα, and supp(A0) ⊆ B ⊆ B∗.

(3) supp(bj) ⊆ Ωj ⊆ B∗j .

(4)∫B∗jbj H dµ = 0 for each H ∈ bp′2(B∗j ). Indeed,∫

B∗j

bj H dµ =

∫Ωj

bj H dµ ∀H ∈ bp′2(B∗j )


and every such H belongs to L∞(Ωj), hence to bq(Ωj) for every q in [1,∞]. Denote by

(·, ·) the inner product in L2(Ωj). Observe that if ϕ is a smooth function with compact

support contained in Bj, and ψ := ϕ−PΩjϕ, then∫Ωj

ψH dµ =

∫Ωj

(I −PΩj

)ϕH dµ

=(ϕ,H

)−(PΩjϕ,H

)=(ϕ,H

)−(ϕ,PΩjH

)= 0,

because ϕ is in L2(Ωj), H belongs to b2(Ωj), PΩj is self-adjoint in L2(Ωj) and is the

identity operator on b2(Ωj).

Now, since Aj is in Lp1(Bj), there exists a sequence ϕjm : m ∈ N of smooth functions

with compact support contained in Bj such that limm→∞∥∥Aj − ϕjm∥∥Lp1 (Bj)

= 0. Then∣∣∣∫Ωj

(I −PΩj

)(Aj − ϕjm)H dµ

∣∣∣≤∥∥(I −PΩj

)(Aj − ϕjm)

∥∥Lp1 (Ωj)

‖H‖Lp′1 (Ωj)

≤(1 + |||PΩj |||Lp1 (Bj);Lp1 (Ωj)

) ∥∥Aj − ϕjm∥∥Lp1 (Bj)‖H‖

Lp′1 (Ωj)

,

which tends to 0 as m tends to ∞. Consequently∫Ωj

bj H dµ = limm→∞

∫Ωj

(I −PΩj

)ϕjmH dµ = 0.

Observe that, by (3) and (4), bj is a multiple of a X1,p2-atom supported in B∗j . It follows that

b is a multiple of a X1,p2-atom supported in B∗∗. Indeed, if H is any function in bp′2(B∗∗),

then H is in L∞(∪jΩj), hence in Lq(∪jΩj) for each q ∈ [1,∞]. We then have that∫B∗∗

bH dµ =∑j

∫Ωj

bj H dµ = 0.

Since both A0 and b satisfy the cancellation condition on the ball B∗∗, g satisfies such

cancellation condition as well. In particular, by (1) and (2),

a0 :=g

Ck1/p10 αNµ(B∗∗)

is a X1,p2 atom, supported in B∗∗.


We now iterate this procedure, performing the same modified Calderon–Zygmund de-

composition to each bj that appears, at each step, in the bad part. Namely, we claim that

there exists a collection of ballsBjl : jl ∈ Nl, l ∈ N

such that, for each n ≥ 1,

A0 = Ck1/p10 αN

n−1∑l=0

αl∑jl∈Nl

µ(B∗∗jl )ajl +∑jn∈Nn

bjn , (2.3.5)

where α = α(p1, D0) is sufficiently large, C is the constant that appears in Proposition

2.2.7(iii) and

(I) ajl is a X1,p2-atom supported in B∗∗jl , for each l ∈ 0, . . . , n− 1;

(II)⋃jn∈Nn Bjn ⊆

x ∈M : M |A0|p1 (x) >

(αn/2

)p1;

(III) the collectionB∗jl

has the finite overlapping property, with constant N l;

(IV) bjn is supported in B∗jn ;

(V)∫B∗jn

bjn H dµ = 0 for any H ∈ bp′2(B∗jn);

(VI) |bjn(x)| ≤ |A0(x)|+ Ck1/p10 αn1B∗jn (x);

(VII)(

1µ(B∗jn )

∫B∗jn|bjn|

p1 dµ)1/p1

≤ (1 + C)k1/p10 αn.

The proof of the claim is by induction on n. We start by proving that

A0 = Ck1/p10 αNµ(B∗∗0 )a0 +

∑j

bj

is the required decomposition for n = 1. Properties (I), (III), (IV) and (V) have already

been established. Since⋃j

Bj ⊆x : M |A0|p1 (x) > αp1

⊆x : M |A0|p1 (x) > (α/2)p1

,

also property (II) holds. Property (VI) follows from Proposition 2.2.7(iv) and the fact that

Aj ≤ A0. Indeed,

|bj(x)| ≤ |Aj(x)|+∣∣PΩj(Aj)(x)

∣∣ 1B∗j (x)

≤ |A0(x)|+ Cµ(Bj)−1/p1 ‖Aj‖Lp1 (Bj) 1B∗j (x)

≤ |A0(x)|+ C

(µ(B∗∗j )

µ(Bj)

)1/p1(

1

µ(B∗∗j )

∫B∗∗j

|A0|p1 dµ

)1/p1

1B∗j (x)

≤ |A0(x)|+ Ck1/p10 α1B∗j (x).


Finally, (VII) is a consequence of (VI):(1

µ(B∗j )

∫B∗j

|bj|p1 dµ

)1/p1

≤

(1

µ(B∗j )

∫B∗j

|A0|p1 dµ

)1/p1

+ Ck1/p10 α

≤ (1 + C)k1/p10 α.

We now assume that (2.3.5) holds for n and we prove it for n+ 1.

For each jn ∈ Nn, set

Ojn =x ∈M : M |bjn|

p1 (x) > α(n+1)p1.

Similarly as before, if α is sufficiently large, Ojn ⊆ 2B∗jn and M \Ojn is non-empty. Then Ojn

admits a Whitney-type decomposition and there exists a collection of ballsBjn,i : i ∈ N

satisfying

(i)⋃iBjn,i = Ojn ;

(ii) B∗∗jn,i ∩Ocjn 6= ∅;

(iii) for each x ∈M ,∑

i 1B∗jn,i(x) ≤ N .

Let Ωjn,i be an approximate (rBjn,i , 1)-ball with centre cBjn,i and denote by PΩjn,ithe

Bergman projection onto b2(Ωjn,i). Let

Ajn,i :=1Bjn,i∑l 1Bjn,l

bjn .

We write

bjn = gjn +∑i

bjn,i,

where gjn := hjn1Ocjn +∑

i PΩjn,i(Ajn,i) and bjn,i := Ajn,i−PΩjn,i

(Ajn,i). Similarly as before,

these equalities make sense in L1(M), and the following hold:

(1)’ |gjn(x)| ≤ Ck1/p10 αn+1N ;

(2)’ supp(gjn) ⊆ 2B∗jn ⊆ B∗∗jn ;

(3)’ supp(hjn,i) ⊆ Ωjn,i ⊆ B∗jn,i;

(4)’∫B∗jn,i

hjn,iH dµ = 0 for each H ∈ bp′2(B∗jn,i).


The proof of these properties is almost identical to the proof of (1), (2), (3) and (4) above,

and is omitted. Moreover, we deduce that∫B∗∗jn

gjn H dµ = 0 ∀H ∈ bp′2(B∗∗jn ).

Hence

ajn :=gjn

Ck1/p10 αn+1Nµ(B∗∗jn )

is a X1,p2-atom supported in B∗∗jn . We then have that

A0 = Ck1/p10 αN

n−1∑l=0

αl∑jl∈Nl


(gjn +

∑i∈N

bjn,i)

= Ck1/p10 αN

n∑l=0

αl∑jl∈Nl


∑i∈N

bjn,i.

We now prove that this is (2.3.5) for n + 1. Properties (I), (IV) and (V) have already been

proved. Property (III) follows from the facts that the ballsB∗jn : jn ∈ Nn

are Mn-disjoint

and the ballsB∗jn,i : i ∈ N

are M -disjoint. To prove (VI), we use the same argument used

to prove (VI) for n = 1 and the inductive hypothesis. Namely,

|bjn,i(x)| ≤ |bjn(x)|+ |Pjn,i(Ajn,i)(x)| 1B∗jn,i(x)

≤(|bjn(x)|+ Cµ(Bjn,i)

−1/p1‖Ajn,i‖Lp1 (Bjn,i)

)1B∗jn,i(x)

≤(|bjn(x)|+ Ck

1/p10 αn+1

)1B∗jn,i(x)

≤(|A0(x)|+ Ck

1/p10 αn1B∗jn (x) + ck

1/p10 αn+1

)1B∗jn,i(x)

≤ |A0(x)|+ Ck1/p10 αn+1 1B∗jn,i(x).

Similarly as before, (VII) follows easily from (VI). It only remains to prove (II). For, observe

that if x is in Ojn , then by (VI)

αn+1 <(M |bjn|

p1 (x))1/p1

≤(M |A0|p1 (x)

)1/p1 + Ck1/p10 αn.

It follows that, if α > 2Ck1/p10 ,

(M |A0|p1 (x)

)1/p1 > αn+1/2. Therefore⋃jn∈Nn

(⋃i∈N

Bjn,i

)=⋃

jn∈NnOjn

⊆x ∈M : M |A0|p1 (x) >

(αn+1/2

)p1,


and (II) is proved.

Then, (2.3.5) is valid for any n ∈ N.

We now prove that (2.3.5) implies (2.3.3), with∑

j |αj| ≤ D, D indipendent of A.

First, we prove that there exists D = D(p1, D0, α) such that

∞∑n=0

αn∑jn∈Nn

µ(B∗∗jn ) ≤ D. (2.3.6)

Observe that ∑jn∈Nn

µ(B∗∗jn ) ≤ k0

∑jn∈Nn

µ(Bjn)

≤ k0Nnµ( ⋃jn∈Nn

Bjn

)≤ k0N

nµ(x ∈M : M

(|A0|p1

)(x) >

(αn+1/2

)p1)≤ C0k0N

n(2/αn

)p1‖A0‖p1p1 .

Here we have used (II) and the fact that M is of weak type 1. It is then enough to choose

α > N1−p1 to obtain

∞∑n=0

αn∑jn∈Nn

µ(B∗∗jn ) ≤ Ck02p1‖A0‖p1p1∞∑n=0

(Nα1−p1

)n≤ Ck02p1µ(B)

∞∑n=0

(Nα1−p1

)n,

and (2.3.6) follows.

Next, we prove that

A0 = Ck1/p10 αN

∞∑n=0

αn∑jn∈Nn

µ(B∗∗jn )ajn , (2.3.7)

where the equality is to be interpreted in L1(M). For any n, set Hn :=∑

jn∈Nn bjn . We show

that ‖Hn‖1 → 0 as n→∞. We recall that bjn is supported in B∗jn and we observe that, by

(VII), ∫B∗jn

|bjn| dµ ≤( ∫

B∗jn

|bjn|p1 dµ

)1/p′1

≤ k1/p10 αnµ(B∗jn).


Therefore, by the above estimate for∑µ(B∗∗jn ),∫

|Hn| dµ ≤∑jn∈Nn

∫|bjn| dµ

≤ k1/p10 αn

∑jn∈Nn

µ(B∗jn)

≤ C2p1k1+1/p10

(α1−p1N

)n‖A0‖p1p1 ,

which tends to 0 as n → ∞, because α1−p1N < 1. This implies (2.3.7) and concludes the

proof of the theorem.

2.4 Calderon–Zygmund decomposition and interpola-

tion

In this section we will prove that, under the assumptions that M is doubling and admits

a relative Faber–Krahn inequality (2.1.2), Lp(M) is an interpolation space between X1(M)

and L∞(M). We recall that M is then a space of homogeneous type in the sense of Coifman

and Weiss. Our interpolation result will be a consequence of a variant of the classical

Calderon–Zygmund decomposition and an argument of J.-L. Journe [J]. Note that this

result is essentially known, though the proofs available in the literature are considerably

involved (see, for instance, [AMR, HLMMY]). It is fair to say, however, that the proof in

[AMR] applies also to Hardy spaces of differential forms, and we do not know whether our

ideas can be pushed to give results also for Hardy spaces of differential forms.

Theorem 2.4.1. Suppose that p is in (1,∞) and that f is in Lp(M). For every positive

number α there exist functions b in X1(M) and g in L∞(M) and a constant C such that

f = b+ g and

(i) |g| ≤ C α;

(ii) ‖b‖X1 ≤ C α |Ωα|, where Ωα denotes the set M (|f |p) > αp, and M is the uncentred

Hardy–Littlewood maximal operator.

Proof. We refer the reader to [CW] for all basic properties of the Hardy–Littlewood maximal

operator on spaces of homogeneous type. We shall use all the properties we need of M

without further reference to [CW].

2.4. CALDERON–ZYGMUND DECOMPOSITION AND INTERPOLATION 57

Denote by Bj a sequence of balls contained in Ωα with the bounded overlapping prop-

erty and such that1

µ(Bj)

∫Bj

|f |p dµ > αp,

but1

µ(B∗j )

∫B∗j

|f |p dµ ≤ αp. (2.4.1)

Here B∗j denotes the ball with the same centre as Bj and twice the radius. Clearly

f 1Ωα =∑j

1Bj∑` 1B`

f.

It will be convenient to set

fj =1Bj∑` 1B`

f.

Note that, by the bounded overlapping property, there exists a positive constant c such that

c f 1Bj ≤ fj ≤ f 1Bj j = 1, 2, . . . .

For every positive integer j, denote by Ωj an approximate (2rBj , 10−2)-ball with centre cBj .

We write

fj = fj −PΩj(fj) + PΩj(fj),

where PΩj(fj) denotes the Bergman projection of fj onto bp(Ωj). Now, define bj := fj −PΩj(fj),

b :=∑j

bj and g := f 1Ωcα +∑j

PΩj(fj). (2.4.2)

It is a standard consequence of the Lebesgue differentiability theorem that |f |p ≤ αp in Ωcα.

Thus, to prove (i) it suffices to show that there exists a constant C such that∥∥∥∑j

PΩj(fj)∥∥∥∞≤ C α.

Since the sequence Ωj has the finite overlapping property, it is enough to show that∥∥PΩj(fj)∥∥∞ ≤ C α,

for some constant C, independent of j. By Proposition 2.2.7 (iv),∥∥PΩj(fj)∥∥∞ ≤

∣∣∣∣∣∣PΩj

∣∣∣∣∣∣Lp(Bj);L∞(Ωj)

∥∥fj∥∥Lp(Bj)

≤ C µ(Bj)−1/p

∥∥fj∥∥Lp(Bj)

≤ C α,


where we have used the doubling condition and (2.4.1) in the last inequality. This concludes

the proof of (i).

Next we prove (ii). We claim that bj is a multiple of a X1,p-atom, and that there exists

a constant C, independent of j, such that∥∥bj∥∥X1 ≤ C µ(B∗j )α. (2.4.3)

Indeed, denote by B∗∗j the ball with centre cBj and radius (2 + 10−1)rBj . Clearly B∗∗j

contains Ωj, and the distance between B∗∗j and ∂Ωj is positive. Furthermore

supp(bj) ⊂ Ωj ⊂ B∗∗j .

First we prove that ∫B∗∗j

bj H dµ = 0 ∀H ∈ bp′(B∗∗j ).

Obviously ∫B∗∗j

bj H dµ =

∫Ωj

bj H dµ ∀H ∈ bp′(B∗∗j )

and every such H belongs to L∞(Ωj), hence to bq(Ωj) for every q in [1,∞]. Denote by (·, ·)the inner product in L2(Ωj). Observe that if ϕ is a smooth function with compact support

contained in Bj, and ψ := ϕ−PΩjϕ, then∫Ωj

ψH dµ =

∫Ωj

(I −PΩj

)ϕH dµ

=(ϕ,H

)−(PΩjϕ,H

)=(ϕ,H

)−(ϕ,PΩjH

)= 0,

because ϕ is in L2(Ωj), H belong to b2(Ωj), PΩj is self-adjoint in L2(Ωj) and is the identity

operator on b2(Ωj).

Now, if fj is any function in Lp(Bj), then there exists a sequence ϕn (which of

course depends on j) of smooth functions with compact support contained in Bj such that

limn→∞∥∥fj − ϕn∥∥Lp(Bj)

= 0. Then∣∣∣∫Ωj

(I −PΩj

)(fj − ϕn)H dµ

∣∣∣ ≤ ∥∥(I −PΩj

)(fj − ϕn)

∥∥Lp(Ωj)

‖H‖Lp′ (Ωj)

≤(1 + |||PΩj |||Lp(Bj);Lp(Ωj)

) ∥∥(fj − ϕn)∥∥Lp(Bj)

‖H‖Lp′ (Ωj),

2.4. CALDERON–ZYGMUND DECOMPOSITION AND INTERPOLATION 59

which tends to 0 as n tends to ∞. Consequently∫Ωj

bj H dµ = limn→∞

∫Ωj

(I −PΩj

)(fj − ϕn)H dµ = 0,

which proves that bj satisfies the cancellation condition on the ball B∗∗j . Hence bj is a multiple

of a X1,p-atom. To prove that bj satisfies (2.4.3), observe that[ ∫B∗∗j

|bj|p dµ]1/p

≤(1 + |||PΩj |||Lp(Bj);Lp(Ωj)

) ∥∥fj∥∥Lp(Bj)

≤ C∥∥f∥∥

Lp(Bj)

≤ C µ(B∗j )1/p[ 1

µ(B∗j )

∫B∗j

|f |p dµ]1/p

≤ C µ(B∗j )1/p α;

the first inequality is a straighforward consequence of the definition of bj, the second follows

from the fact that |||PB∗j|||p is uniformly bounded with respect to j, see Proposition 2.2.7 (iv),

and the fourth from (2.4.1). The above estimate implies that bj/(Cµ(B∗∗j )α

)is a X1,p-atom.

Hence ∥∥bj∥∥X1 ≤ C µ(B∗∗j )α ≤ C µ(B∗j )α;

we have used the doubling condition in the last inequality above. This concludes the proof

of the claim.

Now, the claim implies that ∑j

∥∥bj∥∥X1 ≤ C α∑j

µ(B∗j )

≤ C αµ(Ωα),

by the bounded overlapping property and the doubling condition.

This concludes the proof (ii), and of the theorem.

Theorem 2.4.2. Suppose that T is a linear operator that is bounded from X1(M) to L1(M)

and on Lq(M) for some q in (1,∞]. Then T is bounded on Lp(M) for all p in (1, q).

Proof. We shall prove that T is of weak type (p, p) for all p in (1, q). Then the Marcinkiewicz

interpolation theorem will imply that T is, in fact, bounded on Lp(M) for all p in (1, q).


First we assume that q = ∞. Fix α > 0 and f in Lp(M), and consider a Calderon–

Zygmund decomposition f = b + g at height C ′α as in the previous theorem, where C ′ is

chosen so small that |T g| > α/2 is empty. Then clearly

µ(|T f | > α

)≤ µ

(|T b| > α/2

)+ µ(|T g| > α/2

)≤ 2

α

∫M

∣∣T b∣∣ dµ

≤ 2

α|||T |||X1;L1 ‖b‖X1

≤ C

α|||T |||X1;L1 αµ(Ωα)

≤ C‖f‖pp

αp|||T |||X1;L1 ,

thereby proving that T is of weak type (p, p), as required.

Next, assume that q <∞. Fix α > 0 and f in Lp(M), and consider a Calderon–Zygmund

decomposition f = b + g at height α as in the previous theorem. The measure of the level

set α/2 of T b is estimated exactly as above. To estimate the measure of the level set α/2

of T g we proceed as follows:

µ(|T g| > α/2

)≤ 2q

αq

∫M

∣∣T g∣∣q dµ

≤ 2q

αq|||T |||qq

∫M

∣∣g∣∣p ∣∣g∣∣q−p dµ

≤ C‖f‖pp

αp|||T |||qq;

in the last inequality we have used the fact that |g| ≤ C α and the estimate ‖g‖p ≤ C ‖f‖p.This last inequality follows from the definition of g (see (2.4.2)) and the fact that

∣∣∣∣∣∣PBj

∣∣∣∣∣∣p

is uniformly bounded with respect to j.

Thus T is of weak type (p, p), as required.

2.5 Spectral multipliers

In this section we prove a multiplier result of Mihlin–Hormander type for the Laplace–

Beltrami operator. We emphasize the fact that its proof is fairly simple and avoids using the

theory of singular integral operators. After we completed the proof of this result, X.T. Duong

and L. Yan [DY1] proved a more general result concerning estimates for spectral multipliers

2.5. SPECTRAL MULTIPLIERS 61

for operators satisfying a Davies–Gaffney inequality. In fact, when restricted to spectral

multipliers of the Laplace–Beltrami operator on manifolds with the doubling property and

the relative Faber–Krahn inequality, their result is a strict relative of ours. However, we

believe that our proof is much simpler than that of Duong and Yan, and emphasizes the role

played by the geometric assumptions on M .

For an open bounded set Ω, define

DomΩ(L ) := f ∈ Dom(L ) : supp(f) ⊂ Ω.

In addition to L acting on DomΩ(L ), we consider also the Dirichlet Laplacian LΩ on the

set Ω. We shall restrict our attention to bounded open sets Ω so that ∂Ω is smooth. Then

Dom(LΩ) = W 1,20 (Ω) ∩W 2,2(Ω),

and LΩu is the distributional Laplacian of u for u in Dom(LΩ). The following proposition

will be useful in the proof of Theorem 2.5.4. A version thereof for balls of small radius in

Riemannian manifolds with bounded geometry and spectral gap has been recently proved in

[MMV4, Proposition 3.1]. The proof of Proposition 2.5.1 below follows almost verbatim the

proof of [MMV4, Proposition 3.1], and is omitted.

Given a ball Ω, we denote by TΩ the restriction of L to DomΩ(L ).

Proposition 2.5.1. Assume that Ω is a bounded open subset of M such that ∂Ω is smooth.

The following hold:

(i) W 2,20 (Ω) = DomΩ(L ) ⊂ Dom(LΩ) = W 1,2

0 (Ω) ∩W 2,2(Ω);

(ii) LΩ is an extension of TΩ;

(iii) the operator TΩ is an isomorphism between W 2,20 (Ω) and b2(Ω)⊥, and the inverse op-

erator T −1Ω agrees with the restriction of L −1

Ω to b2(Ω)⊥;

(iv) there exists a constant C, independent of Ω, such that

∥∥T −1Ω f

∥∥2≤ C

λ1(Ω)

∥∥f∥∥L2(Ω)

∀f ∈ b2(Ω)⊥,

where λ1(Ω) denotes the first eigenvalue of the Dirichlet Laplacian LΩ.


The last ingredient of the proof of Theorem 2.5.4 is a lemma, whose statement requires

more notation. For every ν in (−1/2,∞), denote by Jν the Bessel function of the first

kind and of order ν. For notational convenience, we shall denote the operator√

L by

D . Note that for every ν in [−1/2,∞) the function λ 7→ Jν(tλ) is even and entire of

exponential type t, so that the kernel kJν(tD) of the operator Jν(tD) is supported in the

set (x, y) ∈M ×M : d(x, y) ≤ t by the property of finite propagation speed.

Lemma 2.5.2. For N large enough, the following estimate holds

supt>0

∣∣∣∣∣∣JN−1/2(tD)∣∣∣∣∣∣

1<∞.

Proof. For notational convenience, in this proof we shall write J instead of JN−1/2. By

Schwarz’s inequality,∣∣∣∣∣∣J (tD)∣∣∣∣∣∣

1= sup

y∈M

∫M

∣∣kJ (tD)(x, y)∣∣ dµ(x)

≤ supy∈M

µ(B(y, t)

)1/2[ ∫

M

∣∣kJ (tD)(x, y)∣∣2 dµ(x)

]1/2

.

(2.5.1)

We write

J (tD) =[J (tD)

(I + t2L

)N/2] (I + t2L

)−N/2,

denote by Bt the operator J (tD)(I + t2L

)N/2, and observe that, by spectral theory and

the asymptotic behaviour of Bessel functions,

supt>0

∣∣∣∣∣∣Bt

∣∣∣∣∣∣2

= supλ>0

∣∣J (tλ) (1 + t2λ)N/2∣∣ <∞. (2.5.2)

Denote by kt the kernel of the operator(I + t2L

)−N/2. Notice that

kJ (tD)(x, y) = Bt

[kt(·, y)

](x).

Since Bt is bounded on L2(M), uniformly in t by (2.5.2),[ ∫M

∣∣kJ (tD)(x, y)∣∣2 dµ(x)

]1/2

≤ |||Bt|||2[ ∫

M

∣∣kt(x, y)∣∣2 dµ(x)

]1/2

≤ C[ ∫

M

∣∣kt(x, y)∣∣2 dµ(x)

]1/2

.

Recall the classical subordination formula

kt(x, y) = cN

∫∞0

sNe−s ht2s(x, y)ds

s∀x, y ∈M : x 6= y,


and the following upper estimate for the heat kernel on M , which holds under the assump-

tions (2.1.1) and (2.1.2),

hu(x, y) ≤ C

µ(B(y,

√u)) e−cd(x,y)2/u ∀x, y ∈M ∀u > 0.

By inserting this estimate in the subordination formula above, and using the generalised

Minkowski inequality, we have that[ ∫M

∣∣kt(x, y)∣∣2 dµ(x)

]1/2

≤ C

∫∞0

ds

ssN/2e−s

[ ∫M

e−2cd(x,y)2/(t2s)

µ(B(y, t

√s))2 dµ(x)

]1/2

.

To estimate the inner integral on the right hand side, we write M as the union of the annuli

A`, with ` = 0, 1, 2, . . . , where

A` :=x ∈M : `t

√s ≤ d(x, y) < (`+ 1) t

√s.

Note that ∫A`

e−2cd(x,y)2/(t2s) dµ(x) ≤ µ(B(y, (`+ 1) t

√s))

e−2c`2

≤ µ(B(y, t

√s))D`+1

0 e−2c`2 .

Therefore, by summing these estimates with respect to `, we obtain that[ ∫M

e−2cd(x,y)2/(t2s)

µ(B(y, t

√s))2 dµ(x)

]1/2

≤ C µ(B(y, t

√s))−1/2

,

where C does not depend on y and on t. Thus,

∣∣∣∣∣∣J (tD)∣∣∣∣∣∣

1≤ sup

y∈Mµ(B(y, t)

)1/2[ ∫

M

∣∣kt(x, y)∣∣2 dµ(x)

]1/2

≤ C supy∈M

∫∞0

sN/2e−s[ µ(B(y, t)

)µ(B(y, t

√s)

]1/2 ds

s.

Clearlyµ(B(y, t)

)µ(B(y, t

√s)≤ 1 ∀s ∈ [1,∞).

Moreover, by the doubling property, there exist positive numbers C and ν such that

µ(B(y, t)

)µ(B(y, t

√s)≤ C s−ν/2 ∀s ∈ (0, 1) ∀y ∈M.


By combining the last three estimates, we obtain

∣∣∣∣∣∣J (tD)∣∣∣∣∣∣

1≤ C

[ ∫ 1

0

sN−ν/2 e−sds

s+

∫∞1

sN e−sds

s

]1/2

,

where C does not depend on t. The right hand side is finite (and uniformly bounded with

respect to t), as long as N > ν, as required to conclude the proof of the lemma.

Definition 2.5.3. Suppose that J is a positive integer. The space Mih(J) is the vector

space of all even functions f on R for which there exists a positive constant C such that∣∣Djf(ζ)∣∣ ≤ C |ζ|−j ∀ζ ∈ R \ 0 ∀j ∈ 0, 1, . . . , J. (2.5.3)

We denote by ‖f‖Mih(J) the infimum of all constants C for which (2.5.3) holds.

Theorem 2.5.4. Suppose that J is a sufficiently large positive integer. Then there exists a

constant C such that

|||m(√

L )|||X1;L1 ≤ C ‖m‖Mih(J) ∀m ∈ Mih(J).

Proof. In view of the theory developed in [MMV3] it suffices to prove that

sup∥∥m(D)A

∥∥1

: A is an X1-atom<∞.

Suppose that A is an X1-atom, with support contained in B. Observe that∥∥m(D)A∥∥

1=∥∥14Bm(D)A

∥∥1

+∥∥1(4B)cm(D)A

∥∥1.

We estimate the two summands on the right hand side separately. To estimate the first,

simply observe that, by Schwarz’s inequality, the size condition for A, and the spectral

theorem, ∥∥14Bm(D)A∥∥

1≤ µ(4B)1/2

∣∣∣∣∣∣m(D)∣∣∣∣∣∣

2

∥∥A∥∥2

≤(µ(4B)

µ(B)

)1/2

.

The right hand side is bounded independently of B, because M is doubling.

Thus, to conclude the proof of the theorem it suffices to show that

sup∥∥m(D)A

∥∥L1((4B)c)

<∞, (2.5.4)


where the supremum is taken over all X1-atoms A. Suppose that A is a X1-atom associated

to the ball B. Choose ε > 0 and let Ω be an approximate (rB, ε)-ball with centre cB. Note

that A is in b2(Ω)⊥, for every harmonic function in Ω clearly belongs to b2(B). Then, by

Proposition 2.5.1 (ii) and (iii),

A = TΩT −1Ω A = L T −1

Ω A.

Thus,

m(D)A = m(D)L(T −1

Ω A).

Recall that the support of T −1Ω A is contained in Ω and that∥∥T −1

Ω A∥∥L2(Ω)

≤∣∣∣∣∣∣T −1

Ω

∣∣∣∣∣∣L2(Ω)

∥∥A∥∥L2(B)

≤∣∣∣∣∣∣T −1

Ω

∣∣∣∣∣∣L2(Ω)

µ(B)−1/2,

so that ∥∥T −1Ω A

∥∥L1(Ω)

≤ µ(Ω)1/2∥∥T −1

Ω A∥∥L2(Ω)

≤(µ(Ω)

µ(B)

)1/2 ∣∣∣∣∣∣T −1Ω

∣∣∣∣∣∣L2(Ω)

≤ C

λ1(Ω).

Here we have used the doubling property and Proposition 2.5.1 (iv).

We claim that there exists a constant C such that

‖m(D)f‖L1((4B)c) ≤ C r−2B

∥∥f∥∥L1(Ω)

∀f ∈ L1(Ω). (2.5.5)

Given the claim, we conclude that

‖m(D)A‖L1((4B)c) = ‖m(D)L(T −1

Ω A)‖L1((4B)c)

≤ C r−2B

∥∥T −1Ω A

∥∥L1(B)

≤ C r−2B λ1(B)−1

≤ C,

(2.5.6)

thereby completing the proof of (2.5.4), and of the theorem. Note that the first inequality

above follows from the claim, the second from Proposition 2.5.1 (iv), and the last from

(2.1.2).


Thus, to conclude the proof of the theorem it remains to prove the claim. By Fourier

analysis, the restriction of m(D)L f to (4B)c is given by

m(D)L f =1

2π

∫|t|≥(3−ε)rB

m(t) cos(tD)L f dt.

Denote by φ a smooth, even function on R that vanishes in the complement of the set

t ∈ R : 1/4 ≤ |t| ≤ 4. For a fixed R in (0, 1] and for each positive integer i, denote

by Ei the set t ∈ R : 4i−1rB ≤ |t| ≤ 4i+1rB. Clearly φ1/(4irB) is supported in Ei, and∑∞i=1 φ

1/(4irB) = 1 in R \ (−rB, rB). Define mi(D)L f as follows:

mi(D)L f =1

2π

∫Ei

φ1/(4irB)(t) m(t) cos(tD)L f dt. (2.5.7)

Recall that L = D2, so that, at least formally,

cos(tD)L = − d2

dt2cos(tD).

Thus, by integrating by parts, we see that

mi(D)L f = − 1

2π

∫Ei

d2

dt2[φ1/(4irB) m

](t) cos(tD)f dt. (2.5.8)

Now, recall [MMV1, Lemma 5.1] that for every positive integer k there exists a polynomial

Pk+1 of degree k + 1, without constant term, such that∫∞−∞

f(t) cos(vt) dt =

∫∞−∞

Pk+1(O)f(t) Jk+1/2(tv) dt (2.5.9)

for all functions f such that O`f ∈ L1(R) ∩ C0(R) for all ` in 0, 1, . . . , k + 1. Here O`

denotes the differential operator t` ( d/ dt)` on the real line. Thus,

mi(D)L f = − 1

2π

∫Ei

PN(O)d2

dt2[φ1/(4irB) m

](t) JN−1/2(tD)f dt. (2.5.10)

It is straightforward to check that there exists a constant C such that

∣∣∣PN(O)d2

dt2[φ1/(4irB) m

](t)∣∣∣ ≤ C

t3‖m‖Mih(J) ∀t ∈ Ei. (2.5.11)

2.6. DOUBLING PROPERTY AND SCALED POINCARE INEQUALITY 67

Given (2.5.10) and (2.5.11), we obtain that

‖m(D)f‖L1((4B)c) =∞∑i=1

‖mi(D)f‖L1((4B)c)

≤ 1

2π

∞∑i=1

∫Ei

∣∣∣PN(O)d2

dt2[φ1/(4irB) m

](t)∣∣∣ ∥∥J (tD)f

∥∥1

dt

≤ C∞∑i=1

( ∫Ei

t−3 dt)∥∥f∥∥

L1(Ω)

≤ C r−2B

∥∥f∥∥L1(Ω)

,

thereby proving (2.5.5), and concluding the proof of the claim. We have used Lemma 2.5.2

in the second inequality above.

2.6 Doubling property and scaled Poincare inequality

In this section we prove that, under the additional assumption that M supports a scaled L2

Poincare inequality, the Hardy space H1(M) of Coifman and Weiss and the Hardy-type space

X1(M) coincide. We also show that this may not be true if M does not support a scaled

Poincare inequality. These facts are already known. The equivalence of H1(M) and X1(M)

appears, for instance, in [AMR, HLMMY]. Our proof of this result is quite elementary, and

hinges on the fact that, by a result of A. Grigor’yan [Gr1] and of L. Saloff-Coste [Sa1], if a

manifolds possesses the doubling property and supports a scaled Poincare inequality, then it

supports also a uniform Harnack inequality, hence harmonic functions on M are (uniformly)

Holder regular by the celebrated result of E. De Giorgi. We believe that, though the result

is already known, the proof we give is considerably different from the original one and sheds

more light on the role played by the scaled Poincare inequality.

Furthermore, in [AMR] the authors attribute to A. Hassell the result (to the best of our

knowledge still unpublished) that H1(M) and X1(M) are different when M is the connected

product of two Euclidean spaces and claim that this fact may be proved as an application

of methods developed in [CCH]. We give a direct proof of this result that may be applied

also to manifolds which are not necessarily Euclidean at infinity.

Definition 2.6.1. We say that M supports a scaled Poincare inequality if there exists a


constant P such that ∫B

|f − fB|2 dµ ≤ P r2B

∫B

|∇f |2 dµ (2.6.1)

for every f in C∞(B) and for all geodesic ball B. Here fB denotes the average of f over B

and rB is the radius of B.

Before proving the main result of this section, we recall that A. Grigor’yan and L. Saloff-

Coste proved that (2.1.1) and (2.6.1) are equivalent to a uniform parabolic Harnack principle.

This uniform parabolic Harnack principle implies (but it is not equivalent to, see [HS])

a uniform elliptic Harnack principle, hence the uniform Holder continuity of L -harmonic

functions.

Theorem 2.6.2. Suppose that M possesses the volume doubling property (2.1.1) and ad-

mits the scaled Poincare inequality (2.6.1). Then H1(M) = X1(M), and their norms are

equivalent.

Proof. Clearly a X1-atom A is also an H1-atom. Hence a function F in X1(M) is also in

H1(M) and

‖F‖H1 ≤ ‖F‖X1 ∀F ∈ X1(M). (2.6.2)

Therefore the identity map ι is a continuous injection from X1(M) to H1(M). Thus, to

conclude the proof of the theorem it suffices to show that ι is surjective.

In fact, we shall prove the following claim: there exists a constant C such that for every

H1-atom a there exist a summable sequence of complex numbers cj and a sequence Ajof special atoms such that

a =∑j

cj Aj and∑j

|cj| ≤ C.

Suppose that the support of the atom a is contained in the ball B. Fix a positive number

ε. For j = 1, 2, 3, . . . , we denote by Bj and B′j the balls with centre cB and radii 2j−1rB

and (1 + ε)2j−1rB, respectively, and by Pj the orthogonal projection of L2(Bj) onto b2(Ωj),

where Ωj is an approximate (2j−1rB, ε)-ball with centre cB (see Definition 2.2.5). Note that

Bj ⊂ Ωj ⊂ B′j ⊂ Bj+1

for small ε. Sometimes it will be convenient to write P0 for I , the identity operator from

L2(Bj) to L2(Ωj), and B1 instead of B. Define, for any j,

Aj =Pj−1a−Pja

µ(B′j)1/2 ‖Pj−1a−Pja‖2

.


Clearly the support of Aj is contained in B′j, and

‖Aj‖2 ≤ µ(B′j)−1/2.

Observe also that Pja = Pj(Pj−1a), where we tacitly identify L2(Ωj−1) with the corre-

sponding closed subspace of L2(Ωj). Then we may write∫B′j

[Pj−1a−Pja

]H(x) dµ =

∫Ωj

[Pj−1a−Pja

]H(x) dµ

=

∫Ωj

(I −Pj)(Pj−1a)H dµ

for any H in b2(B′j). The operator I −Pj is the orthogonal projection of L2(Ωj) onto

the orthogonal complement of b2(Ωj) in L2(Ωj). Since I −Pj is a self-adjoint operator in

L2(Ωj), and the restriction of H to Ωj is in the kernel of I −Pj, the last integral vanishes.

Thus, Aj is a special atom with support contained in B′j. At least formally, we may write

a = a−P1a+∞∑j=2

[Pj−1a−Pja

]=∞∑j=1

cj Aj,

where cj = µ(B′j)1/2 ‖Pj−1a−Pja‖2. To conclude the proof of the claim, it suffices to show

that∞∑j=1

|cj| ≤ C,

where C does not depend on the atom a. We denote by Rj the reproducing kernel of b2(Ωj).

Since a has vanishing integral, we may write

Pja(x) =

∫B1

a(y)Rj(x, y) dµ(y)

=

∫B1

a(y)[Rj(x, y)−Rj(x, cB)

]dµ(y).

Moreover, by the reproducing and the symmetry properties of Rj,

‖Rj(·, y)−Rj(·, cB)‖2L2(Ωj)

=(Rj(·, y)−Rj(·, cB), Rj(·, y)−Rj(·, cB)

)= Rj(y, y) +Rj(cB, cB)− 2Rj(y, cB)

≤∣∣Rj(y, y)−Rj(y, cB)

∣∣+∣∣Rj(y, cB)−Rj(cB, cB)

∣∣.


The functionsRj(y, ·) andRj(·, cB) are harmonic in Ωj. Since cB and y belong toB1 ⊆ 2−1Bj,

and a uniform parabolic Harnack inequality holds on balls of M , by De Giorgi’s theorem

(see, for instance, [Sa1]) there exist α in (0, 1) and a constant C, independent of y and j,

such that ∣∣Rj(y, y)−Rj(y, cB)∣∣ ≤ C

( |y − cB|2j−1rB

)αsupz∈2B1

∣∣Rj(y, z)∣∣. (2.6.3)

For j large, both y and z are in 2−1Bj. Therefore, by Proposition 2.2.7 (iii), we can conclude

that there exists a constant C, independent of j, such that

supz∈2B1

∣∣Rj(y, z)∣∣ ≤ C µ(Bj)

−1,

so that, by (2.6.3), ∣∣Rj(y, y)−Rj(y, cB)∣∣ ≤ C 2−αj µ(Bj)

−1.

This, the generalised Minkowski inequality and the doubling property imply that

‖Pja‖2 ≤∫B1

|a(y)|∥∥Rj(·, y)−Rj(·, cB)

∥∥2

dµ(y)

≤ C 2−αj µ(Bj)−1/2

∫B1

|a(y)| dµ(y)

≤ C 2−αj µ(Bj)−1/2

= C 2−αj µ(B′j)−1/2

(µ(B′j)

µ(Bj)

)1/2

≤ C 2−αj µ(B′j)−1/2.

(2.6.4)

Therefore

|cj| ≤ C 2−αj,

with C independent of a and j, so that the sequence cj is summable, as required to

conclude the proof of the claim, and of the theorem.

An example of a Riemannian manifold which does not support a uniform Harnack in-

equality is the following. Denote by M the connected product Rn # Rn of two copies of

Rn, when n ≥ 3. The manifold M is obtained by “gluing” smoothly the two copies of

Rn \ B(0, 1), which we denote by C1 and C2. The metric on M agrees with the Euclidean

metric on C1 and C2, and is globally smooth. It is straightforward to check that the scaled

Poincare inequality (2.6.1) fails on M (hence M does not support a uniform Harnack in-

equality, by a well known result of Grigor’yan and Saloff-Coste). Indeed, choose x on the


“cylinder” that joins C1 and C2. Let R > 0 and let fR be a function on M , with support

contained in B(x,R), that is equal to −1 on C1 and +1 on C2. Clearly ∇fR vanishes on C1

and C2, hence its L2-norm does not depend on R, as R tends to infinity. Hence the right

hand side of (2.6.1) grows like R2 as R tends to +∞. However, by carefully choosing the

values of fR on the “cylinder” joining C1 and C2, we may assume that∫B(x,R)

fR dµ = 0.

Then the left hand side of (2.6.1) grows as Rn, as R tends to infinity, whence the scaled

Poincare inequality cannot possibly hold for large values of R.

However, note that Rn#Rn satisfies a relative Faber–Krahn inequality (2.1.2). Indeed, by

the Federer–Fleming theorem (see, for instance, [Ch]), the isoperimetric property of Rn#Rn

implies the Sobolev inequality

‖f‖n/(n−1) ≤ C ‖∇f‖1 ∀f ∈ C∞c (Rn # Rn). (2.6.5)

This, in turn, implies the following estimate for the heat kernel:

ht(x, x) ≤ C t−n/2 ∀x ∈ Rn # Rn, ∀t > 0

(see [Gr2, Corollary 14.23, p. 383]). Finally, the relative Faber–Krahn inequality is a conse-

quence of the last inequality and the doubling property.

Theorem 2.6.3. Denote by M the connected product Rn #Rn. Then ∇L −1/2 is unbounded

from H1(M) to L1(M).

Proof. By (2.6.5), there exists a positive constant C such that

‖∇L −1/2f‖1 ≥ C ‖L −1/2f‖n/(n−1).

If ∇L −1/2 were bounded from H1(M) to L1(M), the following estimate would hold:

‖f‖H1(M) ≥ C ‖L −1/2f‖n/(n−1) ∀f ∈ H1(M).

In particular,

sup ‖L −1/2a‖n/(n−1) <∞,

where the supremum is taken over all H1-atoms a. Denote by k the Schwartz kernel of

L −1/2. We have

L −1/2a(x) =

∫M

k(x, y) a(y) dµ(y) (2.6.6)


Choose a with support contained in B(p,R), where p is on the cylinder which joins C1 and

C2, and R is large. Suppose that a is equal to −1 on C1 and to +1 on C2, and carefully

choose the values of a on the “cylinder”, so that∫Ma dµ = 0. Then a is a multiple of an

H1-atom. Let x be a point in C2, far from the “cylinder”. Then denote by y and y two

points in B(p,R), the first on C2 and the second on C1. By using the estimates for the heat

kernel on M proved by Grygor’yan and Saloff-Coste [GS], it is not hard to prove that

k(x, y) k(x, y).

In fact, we can prove that there exists a positive constant C such that for R large enough∫B

k(x, y) a(y) dµ(y) ≥ c

∫B+

k(x, y) dµ(y),

whereB+ denotesB(p,R)∩C2. Similar estimates show that the function x 7→∫B+k(x, y) dµ(y)

does not belong to Ln/(n−1)(M). Therefore L −1/2a is not in Ln/(n−1)(M).

Part II

A symmetric diffusion semigroup

73

Chapter 3

The semigroup generated by the

operator A

This chapter contains the analysis of the semigroup generated on (Rn, γ−1) by the operator

A . We recall that A is defined by

A f = −1

2∆f − x· ∇f ∀f ∈ C∞c (Rn),

and that γ−1is the measure whose density with respect to Lebesgue measure is πn/2 e|x|2

.

In Section 3.1, we shall prove some preliminary properties of A , produce a formula for the

semigroup e−tA , and investigate its region of holomorphy in Lp(γ−1). In Section 3.2 we shall

briefly study local singular integral operators in our setting. In Section 3.3 we shall prove

that the maximal operator H ∗ associated to the semigroup generated by A is of weak type

1. Finally, Section 3.4 contains the statement of all results we have obtained concerning the

functional calculus for A .

3.1 Background material and preliminary results

Let 1 ≤ p <∞. Denote by Up the operator defined by

Upf = γ−1/p f ∀f ∈ C∞c (Rn) (3.1.1)

(see (0.0.3) for the definition of γ−1/p). Clearly Up extends to an isometry of Lp(γ−1) onto

Lp(λ) and of Lp(λ) onto Lp(γ1). We denote by L the Ornstein–Uhlenbeck operator. A

75

76 CHAPTER 3. THE SEMIGROUP GENERATED BY THE OPERATOR A

straightforward computation shows that

U2A U −12 =

1

2

(−∆ + |·|2 + nI

)and

U −12 L U2 =

1

2

(−∆ + |·|2 − nI

).

Consequently,

A f =1

2U −1

2

(−∆ + |·|2 + nI

)U2f

=1

2U −1

2

(−∆ + |·|2 − nI + 2nI

)U2f

= U −12 U −1

2 (L + nI )U2U2f ∀f ∈ C∞c (Rn).

It is well known that the Ornstein–Uhlenbeck operator L is essentially self-adjoint. We

abuse the notation and still denote by L its self-adjoint extension, and by Dom(L ) its

domain. Clearly L +nI is also self-adjoint. Since U 22 = U1, which extends to an isometry

of L2(γ−1) onto L2(γ1), the operator

U −11 (L + nI ) U1f ∀f ∈ L2(γ1) : U1f ∈ Dom(L )

is a self-adjoint extension of A , which, with abuse of notation, we still denote by A . Its

domain is

Dom(A ) = f ∈ L2(γ−1) : U−1f ∈ Dom(L ).

Thus,

A f = U −11 (L + nI ) U1f ∀f ∈ Dom(A ). (3.1.2)

The spectral resolution of the identity of L is

L =∞∑j=0

jPj,

where Pj denotes the orthogonal projection onto the linear span of the Hermite polynomials

of degree j in n variables. Therefore, the spectral resolution of A is

A =∞∑k=0

(k + n) Ek, (3.1.3)

where Ek = U −11 PkU1. Thus, Ek is the orthogonal projection of L2(γ−1) onto the linear

span of the functions of the form γ1 p, where p is a Hermite polynomial of degree j in n

variables.

3.1. BACKGROUND MATERIAL AND PRELIMINARY RESULTS 77

By spectral theory,

e−tA = U −11 e−t(L +nI ) U1f ∀f ∈ L2(γ−1).

We shall denote by Htt≥0 the semigroup e−tA on L2(γ−1), and by Mtt≥0 the semigroup

e−tL on L2(γ). We recall Mehler’s formula for the Ornstein–Uhlenbeck semigroup. For

t > 0,

Mtf(x) =

∫mt(x, y)f(y) dγ(y), (3.1.4)

where

mt(x, y) =1

(1− e−2t)n/2exp

( |x+ y|2

2(et + 1)− |x− y|

2

2(et − 1)

)∀x, y ∈ Rn ∀t > 0.

It is known that Mtt≥0 is a symmetric diffusion semigroup on (Rn, γ1); we refer to [B] and

the references therein for more on the Ornstein–Uhlenbeck semigroup.

Theorem 3.1.1. For any test function f

Htf(x) =

∫ht(x, y)f(y) dγ−1(y) if t > 0

f(x) if t = 0,(3.1.5)

where

ht(x, y) =e−tn

πn(1− e−2t)n/2exp

(− |x+ y|2

2(1 + e−t)− |x− y|2

2(1− e−t)

). (3.1.6)

Moreover, Htt≥0 is a symmetric diffusion semigroup on (Rn, γ−1).

Proof. To prove (3.1.5), notice that

Htf(x) = U −11 e−t(L +nI )U1f(x)

=e−tn

(1− e−2t)n/2e−|x|

2

∫exp

( |x+ y|2

2(et + 1)− |x− y|

2

2(et − 1)

)e|y|

2

f(y) dγ(y)

=e−tn

πn(1− e−2t)n/2

∫exp

( |x+ y|2

2(et + 1)− |x− y|

2

2(et − 1)− |x|2 − |y|2

)f(y) dγ−1(y)

=e−tn

πn(1− e−2t)n/2

∫exp

(− |x+ y|2

2(1 + e−t)− |x− y|2

2(1− e−t)

)f(y) dγ−1(y),

as required.


We already know that e−tA is symmetric. It is also positivity preserving, for its kernel is

positive. We also note that, for f ∈ L1(γ−1)∩L2(γ−1),

∫f(y) dγ−1(y) =

e−tn

πn/2(1− e−2t)n/2

∫f(y)e|y|

2

dy

∫exp

(− |e

−tx− y|2

1− e−2t

)dx

=e−tn

πn/2(1− e−2t)n/2

∫ ∫exp

(− |x− e−ty|2

1− e−2t+ |x|2

)f(y) dy dx

=

∫Htf(x) dγ−1(x).

Thus, Ht is contractive on L1(γ−1). Hence, by interpolation and duality, Ht is contractive

on Lp(γ−1) for 1 < p <∞. Finally,

Ht 1 = 1 ∀t > 0,

i.e., Ht is markovian, and it is contractive also on L∞(γ−1).

Remark 3.1.2. A noteworthy consequence of (3.1.6) is that the semigroup Ht satisfies

|||Ht|||1;∞ = supx,y∈Rn

ht(x, y)

=e−tn

πn(1− e−2t)n/2∀t > 0,

(3.1.7)

where |||Ht|||1;∞ denotes the operator norm of Ht from L1(γ−1) to L∞(γ−1). Thus, the

semigroup Ht is ultracontractive. By interpolation, Ht is also hypercontractive, i.e., Ht

is bounded from Lp(γ−1) to Lp′(γ−1), for p ∈ (1, 2). We recall that the Gaussian semigroup

Mt is not ultracontractive, and that Mt is bounded from Lp(γ1) to Lp′(γ1) if and only if

t ≥ 12

log(p′−1p−1

), by a well known theorem of E. Nelson [Nel, Theorem 2].

By spectral theory, the map t 7→ Ht from [0,∞) to the space of bounded operators

on L2(γ−1) is continuous with respect to the strong operator topology of L2(γ−1), and it

extends to a continuous map from the right half plane z : Re(z) ≥ 0 to the space of

bounded operators on L2(γ−1), which is analytic in z : Re(z) > 0. Correspondingly, the

kernel t 7→ ht has an analytic continuation to a distribution-valued function z 7→ hz, which

is continuous in Re z ≥ 0 and holomorphic in Re z > 0. Clearly hz is the kernel of Hz.

An interesting question is to determine for which z in the right half plane the operator Hz

extends to a bounded operator on Lp(γ−1), for some 1 ≤ p ≤ ∞. Since

Hzf(x) = Hzf(x), (3.1.8)

Hz is bounded on Lp(γ−1) if and only if Hz is. Moreover, since∫Hzf g dγ−1 =

∫f Hzg dγ−1,


we may conclude that Hz is bounded on Lp(γ−1) if and only if it is bounded on Lp′(γ−1),

where 1/p+ (1, 1)/p′ = 1, and |||Hz|||Lp(γ−1) = |||Hz|||Lp′ (γ−1).

The final part of this section is dedicated to proving that Hz extends to a bounded

operator on Lp(γ−1) if and only if z belongs to the set Ep, defined by

Ep = x+ iy ∈ C : |sin y| ≤ (tanφp) sinhx, (3.1.9)

where φp := arccos(2/p−1). We recall that the set Ep is the region of boundedness on Lp(γ)

for the analytic continuation of the Ornstein–Uhlenbeck semigroup Mt [E, Theorem 1.1].

By looking at the spectral resolution of the identity of A , it is clear that

Hz+iπf(x) = Hzf(−x).

Thus, Ep is a closed iπ–periodic subset of the half–plane Re z ≥ 0, symmetric with respect

to the x-axis, and Ep = Ep′ . So, to investigate the boundedness properties of the semigroup

Hz, we may restrict the parameter z to the set

Fp = z ∈ Ep : 0 ≤ Im z ≤ π

2, (3.1.10)

and we may consider only the case when 1 ≤ p ≤ 2.

The proofs of most of the results below concerning A and Hz make use of the following

change of variables, which was introduced in [GMMST1]. Define the C∪∞-valued function

τ on the set ξ ∈ C : |ξ| ≤ 1, |argξ| ≤ π2 to be as

τ(ξ) =

log 1+ξ

1−ξ if ξ 6= 1,

∞ if ξ = 1.(3.1.11)

It is not difficult to see that τ maps its domain onto the halfstrip

z ∈ C : Re z ≥ 0, |Im z| ≤ π

2 ∪ ∞,

the interval (0, 1) onto (0,∞) and the sector Sφp = ξ ∈ C : |ξ| ≤ 1, 0 ≤ arg(ξ) ≤ φp onto

the set Fp ∪ ∞. It is straightforward to check that, if ξ 6= 1, then

hτ(ξ)(x, y) =(1− ξ)n

πn(4ξ)n/2exp

(− |x|

2 + |y|2

2− 1

4

(ξ |x+ y|2 +

1

ξ|x− y|2

)). (3.1.12)

We also set h∞(x, y) := 1 for all x, y.


Theorem 3.1.3. Let 1 ≤ p ≤ 2. Then Hz is bounded on Lp(γ−1) if and only if z ∈ Ep, and

in this case it is a contraction.

Proof. The case where p = 2 follows directly from spectral theory. Thus, we may assume

that 1 ≤ p < 2.

We observe that Hτ(ξ) is bounded on Lp(γ−1) if and only if H λτ(ξ) = UpHτ(ξ)U

−1p is

bounded on Lp(λ). The kernel of the latter operator with respect to the Lebesgue measure

is of Gaussian type and is given by

hλτ(ξ)(x, y) =(1− ξ)n

πn/2(4ξ)n/2exp

(− |x|

2 + |y|2

2− 1

4

(ξ |x+ y|2 +

1

ξ|x− y|2

)+|x|2

p+|y|2

p′

)=

(1− ξ)n

πn/2(4ξ)n/2exp

(− β(ξ) |x|2 − ε(ξ) |y|2 + 2δ(ξ)x · y

),

where

β(ξ) =1

2+

1

4

(ξ +

1

ξ

)− 1

p, ε(ξ) =

1

2+

1

4

(ξ +

1

ξ

)− 1

p′, δ(ξ) =

1

4

(1

ξ− ξ).

We rewrite the condition ξ ∈ Sφp in terms of the coefficients β, ε and δ. It is easy to see

that, for |ξ| ≤ 1, |arg ξ| ≤ φp if and only if

(Re δ(ξ))2 ≤ (Re β(ξ)) (Re ε(ξ)). (3.1.13)

Indeed,

(Re δ(ξ))2 − (Re β(ξ))(Re ε(ξ))

=(Re ξ

4

( 1

|ξ|2− 1))2

−(Re ξ

4

(1 +

1

|ξ|2)

+1

2− 1

p

)(Re ξ

4

(1 +

1

|ξ|2)− 1

2+

1

p

)= −(Re ξ)2

4 |ξ|2+(1

p− 1

2

)2

=1

4

(− (Re ξ)2

(Re ξ)2 + (Im ξ)2+

(2− p)2

p2

),

so that (Re δ(ξ))2 ≤ (Re β(ξ))(Re ε(ξ)) if and only if∣∣∣∣Im ξ

Re ξ

∣∣∣∣ ≤ 2√p− 1

2− p= tanφp.

We prove that, for Re ξ ≥ 0 and Im ξ ≥ 0, H λτ(ξ) is bounded on Lp(λ) if and only if

(3.1.13) holds.


First we prove that if (3.1.13) holds, then H λτ(ξ) is bounded on Lp(λ). We observe that

H λτ(ξ) =

( πγ∗δ(ξ)

)n/2 (1− ξ)n

πn/2(4ξ)n/2Mβ(ξ)−γ∗δ(ξ) Tγ∗ e−γ

∗/(4δ(ξ)) ∆ Mε(ξ)−δ(ξ)/γ∗

=(γ∗)n/2(1− ξ)n/2

(1 + ξ)n/2Mβ(ξ)−γ∗δ(ξ) Tγ∗ e−γ

∗/(4δ(ξ)) ∆ Mε(ξ)−δ(ξ)/γ∗ ,

(3.1.14)

where γ∗ > 0 is a constant which will be chosen later. Here Mα, for α ∈ C, denotes the

multiplication operator given by

Mαf(x) = e−α|x|2

f(x),

Tγ∗ , for γ∗ > 0, the dilation operator

Tγ∗f(x) = f(γ∗x),

and e−z∆ : Re z ≥ 0 the heat semigroup on (Rn, λ), defined by

e−z∆f(x) =1

(4πz)n/2

∫exp

(− |x− y|24z

)f(y) dy.

A straightforward calculation shows that

exp(− β(ξ) |x|2 − ε(ξ) |y|2 + 2δ(ξ)x · y

)= exp

((−β(ξ) + γ∗δ(ξ)) |x|2

)exp

(− δ(ξ) |γ∗x− y|2 /γ∗

)exp

((−ε(ξ) + δ(ξ)/γ∗) |y|2

),

so that (3.1.14) holds. Now, write ξ = teiφ, with t ∈ (0, 1] and 0 ≤ φ ≤ φp. We shall treat

the two cases Re δ(ξ) = 0 or Re δ(ξ) > 0 separately.

Suppose first that Re δ(ξ) > 0, i.e., t 6= 1. Then Re(γ∗/(4δ(ξ))

)> 0, hence e−γ

∗/(4δ(ξ)) ∆

is bounded on Lp(λ). The operator Mβ(ξ)−γ∗δ(ξ) is bounded on Lp(λ) if and only if

Re β(ξ)− γ∗Re δ(ξ) ≥ 0,

whereas Mε(ξ)−δ(ξ)/γ∗ is bounded on Lp(λ) if and only if

Re ε(ξ)− Re δ(ξ)/γ∗ ≥ 0.

Therefore the two operators are simultaneously bounded if and only if

Re δ(ξ)

Re ε(ξ)≤ γ∗ ≤ Re β(ξ)

Re δ(ξ); (3.1.15)


since ξ ∈ Sφp implies (3.1.13), such a positive γ∗ exists. Then H λτ(ξ) is composition of

bounded operators, hence it is bounded on Lp(λ), and

∣∣∣∣∣∣H λτ(ξ)

∣∣∣∣∣∣Lp(λ)

≤∣∣∣1− ξ1 + ξ

∣∣∣n/2 (γ∗)n/2

(γ∗)n/p. (3.1.16)

Observe that for all ξ we are considering, we have |(1− ξ)/(1 + ξ)| ≤ 1. We need to show

that for every such ξ we may choose γ∗ such that the right hand side of the inequality above

is less than or equal to 1. Note that

Re δ(ξ)

Re ε(ξ)=

cosφ4

(1/t− t)cosφ

4(1/t+ t) + cosφp

2

< 1 ∀t ∈ (0, 1] ∀φ ∈ [0, φp],

and thatRe β(ξ)

Re δ(ξ)=

cosφ4

(1/t+ t)− cosφp2

cosφ4

(1/t− t)≥ 1

if and only if t ≥ cosφp/ cosφ. Therefore, if ξ is such that t ≥ cosφp/ cosφ, then there exists

γ∗ ≥ 1, satisfying (3.1.15), so that, by (3.1.16),∣∣∣∣∣∣H λτ(ξ)

∣∣∣∣∣∣Lp(λ)

≤ (γ∗)n2−np ≤ 1.

If, instead, ξ is such that t < cosφp/ cosφ, then every γ∗ which satisfies (3.1.15) is smaller

than 1. In this case, we set

γ∗ =Re β(ξ)

Re δ(ξ),

so that ∣∣∣∣∣∣H λτ(ξ)

∣∣∣∣∣∣Lp(λ)

≤∣∣∣1− ξ1 + ξ

∣∣∣n/2 (Re β(ξ)

Re δ(ξ)

)n/2−n/p≤(1 + t2 − 2t cosφ

1 + t2 + 2t cosφ

)n/4( cosφ(1− t2)

cosφ(1 + t2)− 2t cosφp

)n/2=(1 + t2 − 2t cosφ

1 + t2 + 2t cosφ

(1− t2)2

(1 + t2 − 2t cosφp/ cosφ)2

)n/4.

The right hand side is at most 1 if and only if

(1 + t2 − 2t cosφ)(1− t2)2 − (1 + t2 + 2t cosφ)(1 + t2 − 2t cosφp/ cosφ)2 ≤ 0,

which is equivalent to

a(φ)t4 + b(φ)t3 + c(φ)t2 + b(φ)t+ a(φ) ≥ 0, (3.1.17)


where

a(φ) = cosφ− cosφp/ cosφ,

b(φ) = 1− 2 cosφp + cos2 φp/ cos2 φ,

c(φ) = 2 cosφp/ cosφ (cosφp − 1).

It is easy to see that (3.1.17) is satisfied for each t < cosφp/ cosφ. Indeed,

a(φ)t4 + b(φ)t3 + c(φ)t2 + b(φ)t+ a(φ)

= t4 cosφ+ t3cosφpcosφ

(cosφpcosφ

− t)

+ t3(1− cosφp) + t2cosφpcosφ

(2 cosφp − 1)

+ t(1− t2 cosφp) + tcosφpcosφ

(cosφpcosφ

− t)

+ cosφ− t cosφp +cosφpcosφ

(1− t cosφ).

It is clear that all the terms here are nonnegative if cosφp ≥ 1/2. Similarly, by writing

a(φ)t4 + b(φ)t3 + c(φ)t2 + b(φ)t+ a(φ)

= t4 cosφ+ t3cosφpcosφ

(cosφpcosφ

− t)

+ t3(1− 2 cosφp) + 2t2cos2 φpcosφ

+ tcosφpcosφ

(cosφpcosφ

− t)

+cosφpcosφ

(1− t2) + t(1− 2 cosφp) +(

cosφ+cosφpcosφ

),

we see that (3.1.17) is satisfied also if cosφp < 1/2. This concludes the proof of the fact

that, if Re δ(ξ) > 0, then condition (3.1.13) implies the contractivity of H λτ(ξ) on Lp(λ).

We now turn to the case Re δ(ξ) = 0, i.e., t = 1. Write ξ = eiφ, and assume momentarily

that φ < φp. Then ∣∣exp(−β(ξ) |x|2 − ε(ξ) |y|2 + 2δ(ξ)x · y

)∣∣= exp

(−Re β(ξ) |x|2 − Re ε(ξ) |y|2 + 2 Re δ(ξ)x · y

)= exp

(−cosφ− cosφp

2|x|2 − cosφ+ cosφp

2|y|2).

Therefore ∫ ∣∣hλτ(ξ)(x, y)∣∣ dx <∞ and

∫ ∣∣hλτ(ξ)(x, y)∣∣ dy <∞.

Thus, H λτ(ξ) is bounded on L1(λ) and on L∞(λ). By interpolation, H λ

τ(ξ) is bounded on

Lp(λ). Moreover, by Fatou’s lemma, for each 0 < φ < φp,∫ ∣∣∣H λτ(eiφ)f

∣∣∣p dλ ≤ lim inft→1

∫ ∣∣∣H λτ(teiφ)f(x)

∣∣∣p dx.


Since H λτ(teiφ)

is a contraction on Lp(λ) for t < 1, it follows that each H λτ(eiφ)

is a contraction

as well. Similarly, ∫ ∣∣∣H λτ(eiφp )

f(x)∣∣∣p dx ≤ lim inf

φ→φp

∫ ∣∣∣H λτ(eiφ)f(x)

∣∣∣p dx,

and so also H λτ(eiφp )

is a contraction on Lp(λ).

Next we prove that if H λτ(ξ) is bounded on Lp(λ), then (3.1.13) holds. Notice that the

assumption p < 2 implies that

Re ε(ξ) =cosφ

4(t+ 1/t) +

cosφp2

> 0. (3.1.18)

We consider the action on H λτ(ξ) on the family of Gaussian functions gs(x) = es|x|

2

, for s ∈ C.Observe that if Re s ≤ Re ε(ξ)− Re δ(ξ), then

H λτ(ξ)gs(x) =

(1− ξ)n

(4ξ)n/2(ε(ξ)− s)n/2exp

(δ2(ξ)− β(ξ)ε(ξ) + β(ξ)s

ε(ξ)− s|x|2

)∀x ∈ Rn.

If Re s < 0, then gs ∈ Lp(λ). Since of H λτ(ξ) is bounded on Lp(λ),

Reδ2(ξ)− β(ξ)ε(ξ) + β(ξ)s

ε(ξ)− s< 0,

whence

Reδ2(ξ)− β(ξ)ε(ξ) + β(ξ)s

ε(ξ)− s+ Re β(ξ) < Re β(ξ). (3.1.19)

Consider the complex map

M(s) =δ2(ξ)− β(ξ)ε(ξ) + β(ξ)s

ε(ξ)− s+ β(ξ) =

δ2(ξ)

ε(ξ)− s.

Clearly M is a Mobius transformation, hence if maps generalized circles in the extended

complex plane into generalized circles. In particular, M carries the line Re s = 0 into a circle

C passing through the origin.

We claim that there exists a point s1 such that Re s1 = 0 and

ReM(s1) =(Re δ(ξ))2

Re ε(ξ).

Assuming the claim for the moment, we immediately conclude the proof of the theorem by

observing that (3.1.19) implies that ReM(s1) ≤ Re β(ξ), i.e., (Re δ(ξ))2 ≤ (Re β(ξ))(Re ε(ξ))

holds.

3.2. LOCAL CALDERON–ZYGMUND THEORY 85

To prove the claim, consider the point s2 = i Im ε(ξ), which minimizes |ε(ξ)− s| subject

to the constraint Re s = 0. Since

|M(s2)| = |δ2(ξ)||ε(ξ)− s2|

≥ |δ2(ξ)||ε(ξ)− s|

for any s such that Re s = 0, we see that the segment joining 0 and M(s2) is a diameter of

C, so that the point M(s2)/2 is the centre of C. The point (M(s2) + |M(s2)|)/2 lies on C

and has the desired real part, indeed

1

2Re(M(s2) + |M(s2)|) =

1

2

(Re

δ2(ξ)

ε(ξ)− i Im ε(ξ)+

∣∣∣∣ δ2(ξ)

ε(ξ)− i Im ε(ξ)

∣∣∣∣ )=

1

2

((Re δ(ξ))2 − (Im δ(ξ))2

Re ε(ξ)+

(Re δ(ξ))2 + (Im δ(ξ))2

Re ε(ξ)

)=

(Re δ(ξ))2

Re ε(ξ).

The claim is proved.

3.2 Local Calderon–Zygmund theory

The kernels of many operators naturally associated to the operator A are singular integral

operators in suitable neighbourhoods of the diagonal of Rn × Rn, which we now define.

Definition 3.2.1. For every ρ > 0, let

Nρ = (x, y) ∈ Rn × Rn : |x− y| ≤ ρ

1 + |x|+ |y|, (3.2.1)

and denote by Gρ its complementary set. We call Nρ and Gρ the local region and the global

region, respectively.

Definition 3.2.2. A linear operator T, mapping the space of test functions into the space of

measurable functions on Rn, is a local Calderon–Zygmund operator if it satisfies the following

assumptions:

(a) T extends to a bounded operator either on Lq(λ) or on Lq(γ−1) for some q, 1 < q <∞,or is of weak type 1 with respect to λ or to γ−1;


(b) there exists a measurable function K, defined off the diagonal in Rn × Rn, such that

for every test function f

Tf(x) =

∫K(x, y)f(y) dy

for all x outside the support of f ;

(c) the function K satisfies

|K(x, y)| ≤ C

|x− y|n, |∂xK(x, y)|+ |∂yK(x, y)| ≤ C

|x− y|n+1 ,

for all (x, y) in the local region N2, x 6= y.

If ϕ is a smooth function on Rn × Rn such that ϕ(x, y) = 1 for (x, y) ∈ N1, ϕ(x, y) = 0 for

(x, y) /∈ N2 and

|∂xϕ(x, y)|+ |∂yϕ(x, y)| ≤ C

|x− y|for x 6= y, (3.2.2)

we define the local part and the global part of the local Calderon–Zygmund operator T by

Tlocf(x) =

∫K(x, y) ϕ(x, y) dy,

Tglobf(x) = Tf(x)− Tlocf(x).

(3.2.3)

Notice that the kernel of Tloc is supported in the local region N2, so that, for all test functions

f and g such that f ⊗ g is supported in the global region G2,∫Tlocf(x) g(x) dx = 0.

More generally, we say that a linear operator S, mapping C∞c (Rn) into C∞c (Rn)′ is a local

operator if ∫Sf(x) g(x) dx = 0 (3.2.4)

for all test functions f and g such that f ⊗ g is supported in a global region.

In this section we shall prove that if T is a local Calderon–Zygmund operator, then its

local part is bounded both on Lp(λ) and on Lp(γ−1), whenever 1 < p <∞, and it is of weak

type 1 with respect to both measures. Furthermore, we shall prove that, for local operators,

strong and weak boundedness with respect to Lebesgue measure and the measure γ−1 are

equivalent. These results are extensions to our setting of a well established technique, which

has been developed in previous papers on the Ornstein–Uhlenbeck operator. In particular,


we follow closely the treatment in [GMST1]. We shall make use of the following covering

lemma, which appears in [GMST1] (except for (7), whose proof, however, is almost verbatim

as that of the corresponding statement for the Gauss measure).

Lemma 3.2.3. There exists a collection of balls

Bj = B(xj, κ/(1 + |xj|)),

where κ = 1/20, such that

1. the collection Bj : j ∈ N covers Rn;

2. the balls 1/4Bj : j ∈ N are pairwise disjoint;

3. for any A > 0, the collection ABj : j ∈ N has the bounded overlap property, i.e.,

sup∑

j χABj <∞;

4. Bj × 4Bj ⊂ N1 for all j ∈ N;

5. for each ρ > 0 there exists δ > 0 such that Bj × (δBj)c ⊂ Gρ;

6. N1/7 ⊆ ∪j(Bj × 4Bj) ⊆ N1;

7. there exist positive constants C1 and C2 such that for all j ∈ N,

C1 e|xj |2

λ(E) ≤ γ−1(E) ≤ C2 e|xj |2

λ(E) (3.2.5)

for each measurable subset E of δBj (δ is as in 5.).

We shall also need the following lemma, whose proof can be found in [GMST2].

Lemma 3.2.4. Let µ be a nonnegative Borel measure on Rn. Given a sequence of nonnegative

measurable functions fj, let f =∑

j χBjfj, where Bj : j ∈ N is the collection of balls in

Lemma 3.2.3. Then

µx : f(x) > α ≤∑j

µx ∈ Bj : fj(x) >α

M (3.2.6)

for all α > 0, where M = sup∑

j χBj . Moreover,

‖f‖q ≤M

(∑j

∫Bj

|fj|q dµ

)1/q

, (3.2.7)

for 1 ≤ q <∞.


Proposition 3.2.5. Let S be a local operator and p be in (1,∞). Then S is of weak type 1

with respect to λ if and only if it is of weak type 1 with respect to γ−1. The same holds for

the Lp boundedness. Moreover the norms of S on Lp(γ−1) and Lp(λ) are comparable.

Proof. Assume that (3.2.4) holds for all f, g ∈ C∞c (Rn) such that f ⊗ g is supported in the

global region Gρ and that S is of weak type 1 with respect to Lebesgue measure. Let α > 0.

Then, by Lemma 3.2.3, there exists δ > 0 such that Bj × (δBj)c ⊂ Gρ. Since S is local, the

values of Sf on Bj depend only on the values of f on δBj, i.e., χBjSf = χBjS(fχδBj). Then,

by Lemma 3.2.4 and (3.2.5),

γ−1x : |Sf(x)| > α ≤∑j

γ−1x ∈ Bj : |Sf(x)| > α

M

≤ C2

∑j

e|xj |2

λx ∈ Bj :∣∣S(fχδBj)(x)

∣∣ > α

M

≤ C2

∑j

e|xj |2C

α

∫δBj

|f(x)| dλ

≤ C2

C1

∑j

C

α

∫δBj

|f(x)| dγ−1

≤ C

α

∫Rn|φ(x)| dγ−1.

The proofs of the converse and of the second part of the theorem are similar. We omit the

details.

Similarly, one can prove the following result.

Lemma 3.2.6. Assume that T is a linear operator mapping the space of L∞ functions with

compact support into the space of measurable functions on Rn. Define, for any measurable

and locally bounded function f,

T 1f :=∑j

χBj T (χ4Bjf),

where Bj is the covering whose existence is established in Lemma 3.2.3. The following

hold:

(i) if T is of weak type 1 with respect to λ or to γ−1, then T 1 is of weak type 1 with respect

to both measures;


(ii) if T is bounded on Lq(λ) or on Lq(γ−1) for some q ∈ (1,∞), then T 1 is bounded on

Lq(λ) and on Lq(γ−1).

Proposition 3.2.7. Suppose that T is a local Calderon–Zygmund operator. The following

hold:

(i) if 1 ≤ q < ∞ and T is bounded on Lq(γ−1), then the operator Tloc, defined in (3.2.3),

is bounded on Lq(γ−1);

(ii) if T is of weak type 1 with respect to the measure γ−1, then so is the operator Tloc.

A similar statement holds with the Lebesgue measure λ in place of γ−1.

Proof. We shall prove (i). The proof of (ii) follows the same lines, and is omitted. The last

statement of the proposition follows directly from (i), (ii) and Lemma 3.2.6.

Suppose that f is a test function and x is in Bj, where Bj is one of the balls introduced

in Lemma 3.2.3. Then

Tlocf(x) = Tf(x)− Tglobf(x)

= T (fχ4Bj)(x) + T (f(1− χ4Bj))(x)−∫K(x, y)(1− ϕ(x, y))f(y) dy

= T (fχ4Bj)(x) +

∫K(x, y)(ϕ(x, y)− χ4Bj(y))f(y) dy.

Since

Tlocf(x) =

∑j Tlocf(x)χBj(x)∑

j χBj(x),

|Tlocf(x)|

≤∣∣∣∑j

χBj(x)T (fχ4Bj)(x)∣∣∣+

∫∑j

χBj(x) |K(x, y)|∣∣ϕ(x, y)− χ4Bj(y)

∣∣ |f(y)| dy

=∣∣T (1)f(x)

∣∣+ T (2) |f | (x),

(3.2.8)

where T (1) is as in Lemma 3.2.6 and T (2) is the operator with kernel

H(x, y) =∑j

χBj(x) |K(x, y)|∣∣ϕ(x, y)− χ4Bj(y)

∣∣ .By Lemma 3.2.6 and property (a) of Definition 3.2.2, T (1) is bounded on Lq(γ−1). Next, we

prove that T (2) is bounded on Lp(γ−1) for all p in [1,∞). We observe that if (x, y) ∈ N1/7, then


(x, y) ∈ N1 by 6. of Lemma 3.2.3, whence ϕ(x, y)−χ4Bj(y) = 0. Moreover, if (x, y) 6∈ N2 and

x ∈ Bj, then y 6∈ N2, and so again ϕ(x, y)− χ4Bj(y) = 0. Thus H is supported in N2 \N1/7.

Let (x, y) be such that

1

7(1 + |x|+ |y|)≤ |x− y| ≤ 2

1 + |x|+ |y|.

By (c) of Definition 3.2.2,

|K(x, y)| ≤ C

|x− y|n

≤ C (1 + |x|+ |y|)n

≤ C (1 + |y|)n,

because |x| ≤ |x− y|+ |y| ≤ 1 + |y|. Therefore,

|H(x, y)| ≤ C (1 + |y|)n∑J

χBj(x)

≤ C (1 + |y|)n.

By Fubini’s theorem,∫ ∣∣T (2)f(x)∣∣ dγ−1(x) ≤ C

∫ ∫(1 + |y|)nχN2(x, y) |f(y)| dy dγ−1(x)

= C

∫(1 + |y|)n |f(y)| dy

∫x: |x−y|≤2/(1+|x|+|y|)

dγ−1(x).

Since

λ(B(y, 2/(1 + |y|))

)≤ C

(2/(1 + |y|)

)nand

γ−1

(B(y, 2/(1 + |y|))

)≤ C e|y|

2

λ(B(y, 2/(1 + |y|))

),

T (2) is bounded on L1(γ−1), that is∫ ∣∣T (2)f(x)∣∣ dγ−1(x) ≤ C

∫|f(x)| dγ−1(x).

By a similar argument,∣∣T (2)f(x)∣∣ ≤ C

∫y: |x−y|≤2/(1+|x|+|y|)

(1 + |y|)n |f(y)| dγ−1(y)

≤ C1

γ−1(B(x, 2/(1 + |x|)))

∫B(x,2/(1+|x|))

|f(y)| dγ−1(y).

3.3. THE MAXIMAL OPERATOR FOR THE SEMIGROUP HT 91

The boundedness of T (2) on L∞(γ−1) then follows from the boundedness of the Hardy–

Littlewood maximal operator on L∞(γ−1). By interpolation, T (2) is bounded on Lp(γ−1) for

each p ∈ (1,∞). By (3.2.8), this concludes the proof of the theorem.

Theorem 3.2.8. Assume that T is a local Calderon–Zygmund operator. Then Tloc is bounded

on Lp(λ) and on Lp(γ−1) for any 1 < p < ∞, and it is of weak type 1 with respect to both

measures.

Proof. By assumption (a) of Definition 3.2.2, T is either bounded on Lq(λ) or on Lq(γ−1),

or it is of weak type 1 with respect to one of those measures. Then, by Proposition 3.2.7,

Tloc is of weak type 1 and of strong type q with respect to both measures. Moreover, Tloc

is a Calderon–Zygmund operator. Indeed, by Definition 3.2.2 (c) and (3.2.2), its kernel

Kloc = Kϕ satisfies

|Kloc(x, y)| ≤ C

|x− y|n, |∂xKloc(x, y)|+ |∂yKloc(x, y)| ≤ C

|x− y|n+1

for all (x, y) ∈ Rn × Rn, x 6= y.

The conclusion then follows from standard Calderon–Zygmund theory and from Propo-

sition 3.2.5.

3.3 The maximal operator for the semigroup Ht

The aim of this section is to prove weak type 1 estimates for the maximal operator H ∗

associated to the semigroup Htt≥0, defined by

H ∗f(x) = supt>0|Htf(x)| . (3.3.1)

We recall that, by Littlewood–Paley–Stein theory, the maximal operator associated to any

symmetric diffusion semigroup, on any positive measure space (X,µ), is bounded on Lp(µ),

for each p ∈ (1,∞] [St1, Maximal Theorem, page 73], (see also [C]). In general, the maximal

operator may fail to be of weak type 1. In the Gaussian context, the weak type (1, 1)

estimate for the maximal operator associated to the Ornstein–Uhlenbeck semigroup is due to

B. Muckenhoupt [MUu] in the 1-dimensional case and to P. Sjogren [SJ] in higher dimensions.

Later, T. Menarguez, S. Perez and F. Soria [MPS] gave another different proof of the theorem

in higher dimensions. Finally, a simpler proof was obtained by J. Garcia-Cuerva, G. Mauceri.


S. Meda, P. Sjogren and J.L. Torrea [GMMST2], as part of a more general theory concerning

the maximal operator associated to the holomorphic Ornstein–Uhlenbeck semigroup. We

now prove that the approach in [GMMST2] applies also in our context. The strategy is to

decompose the maximal operator into the sum of a global and a local part, and analyze them

separately. We define the local and the global parts of H ∗ by

H ∗locf(x) = sup

t>0

∣∣∣∫ ht(x, y)χN(x, y)f(y) dγ−1(y)∣∣∣,

H ∗globf(x) = sup

t>0

∣∣∣∫ ht(x, y)χG(x, y)f(y) dγ−1(y)∣∣∣,

where we writeN andG instead ofN1 andG1, and where χN and χG denote the characteristic

functions of the sets N and G, respectively. We shall reduce the study of the local part

on Lp(γ−1) to the analogous problem on Lp(λ), where standard Calderon–Zygmund theory

applies, and we shall carefully estimate the kernel in the global region. Then the boundedness

properties of the operator H ∗ will follow from the trivial inequality

H ∗f(x) ≤H ∗locf(x) + H ∗

globf(x).

We recall that, by (3.1.12), for any s ∈ (0, 1)

hτ(s)(x, y) =(1− s)n

πn(4s)n/2exp

(− |x|

2 + |y|2

2− 1

4(s |x+ y|2 +

1

s|x− y|2)

), (3.3.2)

where τ is defined in (3.1.11).

We start by considering H ∗loc. We shall need the following preliminary result.

Lemma 3.3.1. There exists a constant C such that, for every s ∈ (0, 1] and (x, y) ∈ N,

∣∣hτ(s)(x, y)∣∣ ≤ C

e−|y|2

sn/2exp

(− 1

4s|x− y|2

). (3.3.3)

Proof. Clearly,∣∣hτ(s)(x, y)∣∣ ≤ C

sn/2exp

(− |x|

2 + |y|2

2− 1

4(s |x+ y|2 +

1

s|x− y|2)

). (3.3.4)

If (x, y) ∈ N , then |x+ y| |x− y| ≤ |x− y| (|x|+ |y|+ 1) ≤ 1, so that |x|2 ≥ |y|2 − 1. Hence

exp(− |x|

2 + |y|2

2

)≤ C exp(− |y|2).

Then (3.3.3) follows from this, (3.3.4), and the trivial estimate exp(−1/4 s |x+ y|2) ≤ 1.


Theorem 3.3.2. The operator H ∗loc is of weak type 1 and of strong type q for each q ∈ (1,∞].

Proof. Let f ≥ 0. By Lemma 3.3.1,

H ∗locf(x) ≤ C sup

t>0t−n/2

∫exp

( |x− y|24t

)χL(x, y)f(y) dy

= CW ∗f(x),

where W ∗ is the maximal operator associated to the Gauss–Weierstrass semigroup. Therefore

H ∗loc is bounded on Lq(λ) for all q ∈ (1,∞] and of weak type 1 with respect to Lebesgue

measure. Since H ∗loc is local, the same is true with respect to the measure γ−1, by Proposition

3.2.5.

Next, we turn to H ∗glob. We introduce the quadratic form

Qs(x, y) = |(1 + s)y − (1− s)x|2 . (3.3.5)

It is straightforward to check that

s |x+ y|2 +1

s|x− y|2 =

1

sQs(x, y)− 2 |y|2 + 2 |x|2 ,

whence

hτ(s)(x, y) =(1− s)n

πn(4s)n/2exp

(− |x|2 − 1

4sQs(x, y)

). (3.3.6)

Lemma 3.3.3. Denote by θ = θ(x, y) the angle between the two non–zero vectors x and y

(understood to be 0 if n = 1). Let δ > 0. Then there exists a constant C such that, for each

x, y 6= 0 with (x, y) ∈ G,

sups∈(0,1]

(1− s)n

sn/2exp

(− δ

sQs(x, y)

)≤ C[(1 + |x|)n ∧ (|x| sin θ)−n].

Proof. Let (x, y) be in G, with x 6= y. We start by proving the first estimate. We claim that

|x− y| ≥ 1

2(1 + |x|)−1. (3.3.7)

Indeed, if |y| ≥ 1 + |x|, then

|x− y| ≥ |y| − |x| ≥ 1 ≥ 1

2(1 + |x|),


whereas |y| < 1 + |x| implies |x|+ |y| ≤ 1 + 2 |x| , and so

|x− y| ≥ 1

1 + |x|+ |y|≥ 1

2(1 + |x|).

WriteQs(x, y) = |(1 + s)(y − x)− 2sx|2

≥ |(1 + s) |x− y| − 2s |x||2 ,

and note that, if s ≤ 1/(8(1 + |x|)2), then

s |x| ≤ |x|4(1 + |x|)2

≤ 1

4(1 + |x|),

whereas

(1 + s) |x− y| ≥ 1

2(1 + |x|).

Hence

Qs(x, y) ≥ C1

(1 + |x|)2. (3.3.8)

It follows that

sup0<s≤ 1

8(1+|x|)2

(1− s)n

sn/2exp

(− δ

sQs(x, y)

)≤ sup

0<s≤ 18(1+|x|)2

C

sn/2exp

(− δ

s(1 + |x|)2

)≤ C(1 + |x|)n.

Finally, it is enough to majorize the exponential by 1 to be able to deduce that

sup1

8(1+|x|)2<s≤1

(1− s)n

sn/2exp

(− δ

sQs(x, y)

)≤ C

(8(1 + |x|)2

)n/2≤ C(1 + |x|)n.

To prove the second estimate, observe that |(1 + s)y − (1− s)x| ≥ (1−s) |x| sin θ, (the right

hand side is the length of the projection of (1 − s)x on the hyperplane orthogonal to y).

Then

sup0<s≤1

(1− s)n

sn/2exp

(−δsQs(x, y)

)≤ sup

0<s≤1

(1− s)n

sn/2exp

(−δ(1− s)

2

s|x|2 sin2 θ

)≤ C (|x| sin θ)−n,

as required.


We need one more preliminary result, which is contained in the following lemma. Its

proof follows closely the lines of the proof of [GMST1, Theorem 1].

Lemma 3.3.4. The operator T , defined on test functions by

T f(x) = e−|x|2

∫(1 + |x|)n ∧ (|x| sin θ)−nf(y) dγ−1(y), (3.3.9)

extends to an operator of weak type 1 with respect to the measure γ−1.

Proof. Assume that ‖f‖L1(γ−1) = 1 and f ≥ 0, and fix α > 0. Let Eα and ξα denote the level

set x : T f(x) > α and the biggest amongst the solutions of the equation

e−r2

(1 + r)n = α, (3.3.10)

respectively. Then Eα ⊆ B(0, ξα), because |x| ≥ ξα implies

T f(x) ≤ e−|x|2

(1 + |x|)n ≤ α. (3.3.11)

We need only to consider the intersection between Eα and the ring R = ξα/2 ≤ |x| ≤ ξα,because

γ−1

(B(0, ξα/2

))≤ C

∫ ξα/20

eρ2

ρn−1 dρ

≤ C eξ2α/4(ξα/2

)n−2

≤ C eξ2α (1 + ξα)−n

≤ C α−1.

Set E ′α = x′ ∈ Sn−1 : ∃ρ ∈ (ξα/2, ξα) such that ρx′ ∈ Eα and, for x′ ∈ E ′α, denote by r(x′)

the biggest such ρ. Then T f(r(x′)x′) = α by continuity, and this implies

er(x′)2 ∼ 1

α

∫ξnα ∧ (ξα sin θ)−nf(y) dγ−1(y). (3.3.12)

Let dx′ denote the surface measure on Sn−1. We have

γ−1(Eα ∩R) =

∫E′α

dx′∫ r(x′)ξα/2

eρ2

ρn−1 dρ

≤ C

∫E′α

er(x′)2r(x′)n−2 dx′

≤ C

α ξ2α

∫E′α

dx′∫ξ2nα ∧ (sin θ)−nf(y) dγ−1(y).


Now, we change the order of integration and observe that∫Sn−1

ξ2nα ∧ (sin θ)−n dx′ ≤ Cξ2

α, (3.3.13)

as required to obtain the estimate for γ−1(Eα).

Theorem 3.3.5. The operator H ∗glob is of weak type 1 and of strong type q for each q ∈

(1,∞].

Proof. The operator H ∗glob is trivially bounded on L∞(γ−1); moreover, it is controlled by

the operator T defined in (3.3.9), whence it follows from formula (3.3.6), Lemma 3.3.3 and

Lemma 3.3.4 that T is also of weak type 1. The boundedness of H ∗glob on Lp(γ−1) follows

from these estimates by interpolation.

Theorem 3.3.6. The operator H ∗, defined in (3.3.1), is of weak type 1 and of strong type

q for each q ∈ (1,∞].

Proof. The result follows directly from Theorem 3.3.2 and Theorem 3.3.5.

3.4 Multiplier operators for A

In this section we summarize some of the results we have obtained concerning spectral

multipliers associated to the operator A . As explained in the introduction, we do not

provide details of the proofs.

Recall formula (3.1.3), which we rewrite as

A =∞∑k=n

k Ek−n, (3.4.1)

where Ek = U −11 PkU1. Given a bounded sequence M : n, n + 1, . . . → C, we define the

spectral multiplier operator associated to the spectral multiplier M by

M(A )f =∞∑j=n

M(j)Ejf ∀f ∈ L2(γ−1). (3.4.2)

Assume now that M is of Laplace transform type, i.e., it is of the form

M(j) = j

∫∞0

φ(t) e−tj dt ∀j ≥ n, (3.4.3)

3.4. MULTIPLIER OPERATORS FOR A 97

for some bounded measurable function φ defined on (0,∞). Then we say that the operator

M(A ) associated to M is a multiplier operator of Laplace transform type. Since M is

bounded on the spectrum of A , M(A ) is bounded on L2(γ−1).

The following result is the counterpart for A of a well known theorem concerning the

Ornstein–Uhlenbeck operator [GMST2].

Theorem 3.4.1. If the function M is of Laplace transform type, then the multiplier operator

M(A ) is of weak type 1 with respect to the measure γ−1.

Next, we state a result concerning bounded holomorphic functional calculus of order 1/2

in Lp(γ−1), for 1 < p <∞. We need some notation. For any ψ ∈ (0, π), we denote byH∞(Sψ)

the space of bounded holomorphic functions on the open sector Sψ = z ∈ C : |arg z| < ψ. It

is known that every function M in H∞(Sψ) admits a bounded extension to Sψ, also denoted

by M.

Definition 3.4.2. Assume that J is a nonnegative integer and that ψ ∈ (0, π/2). Define

H∞(Sψ; J) as the Banach space of all the functions M in H∞(Sψ) for which there exists a

constant C such that

supR>0

∫ 2R

R

∣∣λjDjM(e±iψλ)∣∣2 dλ

λ≤ C2 ∀j ∈ 0, 1, . . . , J, (3.4.4)

endowed with the norm

‖M‖ψ;J = inf C : (3.4.4) holds.

Condition (3.4.4) is called a Hormander condition of order J .

For any b in R, denote by τb the translation operator

τb(x) = x− b ∀x ∈ R.

Our main results are the following theorems. Set φ∗p = arcsin |2/p− 1| .

Theorem 3.4.3. If 1 < p <∞ and u ∈ R, then∣∣∣∣∣∣A iu∣∣∣∣∣∣Lp(γ−1)

eφ∗p|u| as u tends to ∞.

.


Theorem 3.4.4. Suppose that a is a positive number, and that 1 < p < ∞, p 6= 2. Let

M : n, n + 1, . . . → C be a bounded sequence and assume that there exists a bounded

holomorphic function M in a+ Sφ∗p such that

M(k) = M(k) ∀k ≥ n.

Then the following hold:

(i) if τ−aM ∈ H∞(Sφ∗p ; 1/2), then M(A ) extends to a bounded operator on Lp(γ−1), hence

on Lq(γ−1) for all q such that |1/q − 1/2| ≤ |1/p− 1/2| . Furthermore,

|||M(A )|||Lp(γ−1) ≤ C(‖M‖∞ + ‖M‖φ∗p;J

),

where C does not depend on M ;

(ii) if τ−aM ∈ H∞(Sφ∗p) and |1/q − 1/2| < |1/p− 1/2| , then M(A ) extends to a bounded

operator on Lq(γ−1).

We may assume that a > n. Observe that

M(A ) =∑

k≤ a−n

M(k + n) Ek +∑

k>a−n

M(k + n) Ek

=∑

k≤ a−n

M(k + n) Ek +∑

k>a−n

M(k + n− a+ a) Ek

=∑

k≤ a−n

M(k + n) Ek +∞∑j=1

(τ−aM

)(j) Ej+a−n.

(3.4.5)

Observe that τ−aM is in H∞(Sφ∗p ; J). Denote by Sa the operator, spectrally defined on

L2(γ−1) by

Saf =∞∑j=1

Ej+a−nf.

Thus,

M(A ) =∑

k≤ a−n

M(k + n) Ek +(τaM

)(A )(Saf).

Since the projection Ek extend to operators bounded on Lp(γ−1), Sa extends to a bounded

operator on Lp(γ−1).

Now we briefly discuss some necessary conditions for M(A ) to be an Lp(γ−1) spectral

multiplier. Our approach follows the same lines as [HMM].

3.4. MULTIPLIER OPERATORS FOR A 99

Following Hebisch, Mauceri and Meda [HMM], we introduce the class of Lp(γ−1) uniform

spectral multipliers of A , as the space of those multipliers M satisfying

supt>0|||M(tL )|||Lp(γ−1) <∞.

We have proved the following.

Theorem 3.4.5. Assume that M : [0,∞)→ C is bounded, and continuous on (0,∞). The

following hold:

(i) if p is in (1,∞) \ 2 and M is a Lp(γ−1) uniform spectral multiplier of A , then M

extends to a bounded holomorphic function in the sector Sφ∗p and

‖M‖H∞(Sφ∗p ) ≤ supt>0|||M(tA )|||Lp(γ−1) ;

(ii) supt>0 |||M(tA )|||L1(γ−1) <∞ if and only if M extends to a bounded holomorphic func-

tion in the half plane Sπ/2, and M(−2i·) is the Fourier transform of a finite measure

µM on R, supported in [0,∞); furthermore,

‖µM‖M(R) = supt>0|||M(tA )|||L1(γ−1) .

Bibliography

[A] N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68, 1950,

337–404.

[Au] T. Aubin, A Course in Differential Geometry Graduate Studies in Mathematics,

Vol. 27, American Mathematical Society, 2001.

[ABR] S. Axler, P. Bourdon and W. Ramey, Harmonic Function Theory, Springer–

Verlag, Berlin Heidelberg New York, 1992.

[AMR] P. Auscher, A. McIntosh and E. Russ, Hardy spaces of differential forms on

Riemannian manifolds, J. Geom. Anal. 18, 2008, 192–248.

[B] D. Bakry, L’hypercontractivite et son utilisation en theorie des semi–groupes,

in “Lectures on Probability Theory” (D. Bakry, R.D. Gill, and S.A. Molchanov,

Eds.), Lectures Notes in Mathematics 1581, Springer-Verlag, New York/Berlin,

1994, 1–114.

[Be] S. Bergman, The Kernel Function and Conformal Mapping, American Mathe-

matical Society, 1950.

[BP] J.H. Bramble and L.E. Payne, Mean value theorems for polyharmonic functions,

Amer. Math. Monthly 73, 1966, 124–127.

[C] M.G. Cowling, Harmonic analysis on semigroups, Ann. of Math. 117, 1983,

267–283.

[CCH] G. Carron, T. Coulhon and A. Hassel, Riesz transform and Lp-cohomology for

manifolds with Euclidean ends, Duke Math. J. 133, 2006, 59–93.

101

102 BIBLIOGRAPHY

[Ch] I. Chavel, Isoperimetric inequalities. Differential geometric and analytic per-

spectives., vol. 145 of Cambridge Tracts in Mathematics, Cambridge University

Press, Cambridge, 2001.

[CKY] B.R. Choe, H. Koo and H. Yi, Derivatives of harmonic Bergman and Bloch

functions on the ball, J. Math. Anal. Appl. 260, 2001, 100–123.

[CC] C.V. Coffman and J. Cohen, The duals of harmonic Bergman spaces, Proc.

Amer. Math. Soc. 110, 1990, 697–704.

[Co] R.R. Coifman, A real-variable characterization of Hp, Studia Math. 51, 1974,

269–274.

[CW] R.R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in anal-

ysis, Bull. Amer. Math. Soc. 83, 1977, 569–645.

[CD] T. Coulhon and X.T. Duong, Riesz transforms for 1 ≤ p ≤ 2, Trans. Amer.

Math. Soc. 351, 1999, 1151–1169.

[DA] E.B. Davies, Heat Kernels and Spectral Theory, Cambridge Tracts in Math. 92,

Cambridge University Press, Cambridge, 1989.

[DY1] X.T. Duong and L.X. Yan, Spectral multipliers for Hardy spaces associated

to non-negative self-adjoint operators satisfying Davies-Gaffney estimates, J.

Math. Soc. Japan 63, 2011, 295–319.

[E] J.B. Epperson, The hypercontractive approach to exactly bounding an operator

with complex Gaussian kernel, J. Funct. Anal. 87, 1990, 1–30.

[FeS] C. Fefferman and E.M. Stein, Hp spaces of several variables, Acta Math. 179,

1972, 137–193.

[G] A. Garsia, Topics in almost everywhere convergence, Lectures in Advanced Math-

ematics 4, Markham, Chicago, 1970.

[Ga] M.P. Gaffney, A special Stokes’s theorem for complete Riemannian manifolds,

Ann. of Math. 60, 1954, 140–145.

BIBLIOGRAPHY 103

[GMMST1] J. Garcia-Cuerva, G. Mauceri, S. Meda, P. Sjogren and J.L. Torrea, Functional

calculus for the Ornstein–Uhlenbeck operator, J. Funct. Anal. 183, 2001, 413–

450.

[GMMST2] J. Garcia-Cuerva, G. Mauceri, S. Meda, P. Sjogren and J.L. Torrea, Maximal

operators for the holomorphic Ornstein–Uhlenbeck semigroup, J. London Math.

Soc. (2) 67, no. 1, 2003, 219–234.

[GMST1] J. Garcia-Cuerva, G. Mauceri, P. Sjogren and J.L. Torrea, Higher–order Riesz

operators for the Ornstein–Uhlenbeck semigroup, Potential Anal. 10, 1999, 379–

407.

[GMST2] J. Garcia-Cuerva, G. Mauceri, P. Sjogren and J.L. Torrea, Spectral multipliers

for the Ornstein–Uhlenbeck semigroup, J. Anal. Math. 78, 1999, 281–305.

[Gr1] A. Grigor’yan, The heat equation on noncompact Riemannian manifolds, Math-

ematics of the USSR-Sbornik 72, 1992, 47–77.

[Gr2] A. Grigor’yan, Heat Kernel and Analysis on Manifolds, AMS/IP studies in

advanced mathematics 47, American Mathematical Society and International

Press, 2009.

[GS] A. Grigor’yan and L. Saloff-Coste, Heat kernel on connected sums of Riemannian

manifolds Math. Res. Letters 6, 1999, 307–321.

[HMM] W. Hebisch, G. Mauceri and S. Meda, Holomorphy of spectral multipliers of the

Ornstein–Uhlenbeck operator, J. Funct. Anal. 210, 2004, 101–124.

[HS] W. Hebisch and L. Saloff-Coste, On the relation between elliptic and parabolic

Harnack inequalities, Ann. Inst. Fourier 51, 2001, 1437–1481.

[HLMMY] S. Hofmann, G. Lu, D. Mitrea, M. Mitrea and L. Yan, Hardy spaces associated to

non-negative self-adjoint operators satisfying Davies–Gaffney Estimates, Mem.

Amer. Math. Soc. 214, 2011, no. 1007.

[J] J.-L. Journe, Calderon-Zygmund Operators, Pseudo-Differential Operators and

the Cauchy Integral of Calderon, Lecture Notes in Mathematics, 994, Springer

Verlag, 1983.

104 BIBLIOGRAPHY

[KK] H. Kang and H. Koo, Estimates of the harmonic Bergman kernel on smooth

domains, J. Funct. Anal. 185, 2001, 220–239.

[La] R.H. Latter, A decomposition of Hp in terms of atoms, Studia Math. 62, 1978,

92–101.

[LW] Li P. and Wang J., Mean value inequalities, Indiana Univ. Math. J. 48, 1999,

1257–1283.

[LP] V.A Liskevich and M.A. Perelmuter, Analyticity of submarkovian semigroups,

Proc. Amer. Math. Soc. 123, 1995, 1097–1104.

[Ma] S. Malyshev, The Bergman kernel and the Green function, J. Math. Sci. 87,

1997, 3366–3380.

[MMS] G. Mauceri, S. Meda and P. Sjogren, Sharp estimates for the Ornstein-Uhlenbeck

operator, Ann. Sc. Norm. Sup. Pisa III, 2004, 447–480.

[MMV1] G. Mauceri, S. Meda and M. Vallarino, Hardy type spaces on certain noncom-

pact manifolds and applications, J. London Math. Soc. 84, 2011, 243–268

[MMV2] G. Mauceri, S. Meda and M. Vallarino, Atomic decomposition of Hardy type

spaces on certain noncompact manifolds, J. Geom. Anal. 22, 2012, 864–891.

[MMV3] G. Mauceri, S. Meda and M. Vallarino, The dual of Hardy type spaces, Bergman

spaces and the Poisson equation on certain noncompact manifolds, preprint.

[MMV4] G. Mauceri, S. Meda and M. Vallarino, Sharp boundedness results for imaginary

powers and Riesz transforms on certain noncompact manifolds, preprint.

[Me] S. Meda, A general multiplier theorem, Proc. Amer. Math. Soc. 110, 1990,

639–647.

[Me1] S. Meda, Estimates for the Bergman kernel on Riemannian manifolds, in prepa-

ration.

[MPS] T. Menarguez, S. Perez and F. Soria, The Mehler maximal function: a geometric

proof of the weak type 1, J. London Math. Soc. 2 62, 2000, 846–856.

BIBLIOGRAPHY 105

[MUu] B. Muckenhoupt, Poisson integrals for Hermite and Laguerre expansions, Trans.

Amer. Math. Soc. 139, 1969, 231–242.

[NRSW] A. Nagel, J.P. Rosay, E. Stein and S. Wainger, Estimates for the Bergman and

Szego kernels in C2, Ann. of Math. 129, 1989, 113–149.

[Nel] E. Nelson, The free Markov field, J. Funct. Anal. 12, 1973, 211–227.

[Nic] M. Nicolesco, Les fonctions polyharmoniques, Act. Scient. Industr. 331, Her-

mann et Cie Edit., Paris, 1936.

[Piz] P. Pizzetti, Sul significato geometrico del secondo parametro differenziale di una

funzione sopra una superficie qualunque, Rend. Lincei 18, 1909, 309–316.

[R] M. Riesz, Sur les fonctions conjuguees, Math. Zeit. 27, 1927, 218–244.

[Ru] E. Russ, H1–L1 boundedness of Riesz transforms on Riemannian manifolds and

on graphs, Pot. Anal. 14, 2001, 301–330.

[Sa1] L. Saloff-Coste, A note on Poincare, Sobolev and Harnack inequalities, Duke

Math. J., IMRN 2, 1992, 27–38.

[SJ] P. Sjogren, On the maximal function for the Mehler kernel, in Harmonic Analysis

(Cortona, 1982), Springer Lecture Notes in Math. 992, 1983, 73–82.

[St1] E.M. Stein, Topics in Harmonic Analysis Related to the Littlewood–Paley The-

ory, Annals of Math. Studies, No. 63, Princeton N. J., 1970.

[St2] E.M. Stein, Harmonic Analysis. Real variable methods, orthogonality and oscil-

latory integrals, Princeton Math. Series No. 43, Princeton N. J., 1993.

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